Right Triangle Trigonometry

Size: px
Start display at page:

Download "Right Triangle Trigonometry"

Transcription

1 CONDENSED LESSON 1.1 Right Tringle Trigonometr In this lesson ou will lern out the trigonometri rtios ssoited with right tringle use trigonometri rtios to find unknown side lengths in right tringle use trigonometri inverses to find unknown ngle mesures in right tringle Suppose ou fl kite. There is strong wind, so the string is pulled tut. You hve mrked the string, so ou know how muh string hs een let out, nd ou n mesure the ngle the string mkes with the horizontl. You n use trigonometri rtio to find the kite s height. In this lesson ou will lern how. Trigonometr reltes the ngle mesures in right tringles to the side lengths. First, rell tht tringles with the sme ngle mesures re similr, nd so the rtios of orresponding sides re equl. In right tringles, there re speil nmes for the rtios. For n ute ngle A in right tringle, the sine of A is Hpotenuse the rtio of the length of the leg opposite A to the length of the hpotenuse. sin A opposite leg A hpotenuse The osine of A is the rtio of the length of the leg C djent to A to the length of the hpotenuse. os A djent leg hpotenuse This leg is djent to A. The tngent of A is the rtio of the length of the opposite leg to the length of the djent leg. opposite leg tn A djent leg Red Emple A in our ook, nd then red the emple elow. This leg is opposite A. EXAMPLE Find the unknown length,. 1 C 5 A Solution You know the length of the side opposite to the 5 ngle, nd ou wnt to find the length of the hpotenuse. Therefore, ou n use the sine rtio. sin sin 5 (ontinued) Disovering Advned Alger Condensed Lessons CHAPTER Ke Curriulum Press

2 Lesson 1.1 Right Tringle Trigonometr (ontinued) The inverse of trigonometri funtion gives the mesure of the ngle tht hs given rtio. For emple, sin 30 1_, so sin 1 1_ 30. Emple in our ook uses the inverse tngent funtion. Red the emple refull. Investigtion: Steep Steps Red the opening prgrph of the investigtion in our ook. Complete Steps 1 of the investigtion nd then ompre our nswers to those elow. Step 1 First sketh step with the mimum rise nd minimum run. Let the ngle of inlintion e. euse the floor nd the run re oth horizontl (nd thus prllel), the ngle etween the run nd the hpotenuse is lso. You know the lengths of the opposite nd djent sides, so use tngent to solve for. tn tn in. The ngle of inlintion is out 38. Step Two sets of stirs tht will fit oth the ode nd the rule of thum re set with unit run of 11 in. nd unit rise of 6.5 in. nd set with unit run of 11.5 in. nd unit rise of 6 in. The respetive ngles of inlintion for these sets re given tn nd tn An emple of set of stirs tht fits the rule of thum ut does not fit the ode is one with unit rise of 8.75 in. nd unit run of 8.75 in. The ngle of inlintion for this set is given tn Step 3 Refer to the photo nd digrm on pge 68 of our ook.. There re infinitel mn designs possile, ut not ll designs will meet the ode given in Step 1. For emple, stir with unit rise of out 15.6 in. nd unit run of out 1 in. would fit the 0.8 ngle of inlintion ut would not fit the ode, euse the rise is too high.. To find the solution, let the unit run e represented r. Then the unit rise will e represented 17.5 r. To find r, use the tngent rtio. tn r r r r r 17.5 r r 17.5 r r 1.68 in. So the run is 1.68 in. nd the rise is in. Step Use the tngent funtion nd let e the ngle of inlintion. Using tn 1 16, tn nd using tn 1 0, tn So the ngle should e etween.86 nd in. 176 CHAPTER 1 Disovering Advned Alger Condensed Lessons 010 Kendll Hunt Pulishing

3 CONDENSED LESSON 1. The Lw of Sines In this lesson ou will disover nd ppl the Lw of Sines, whih desries reltionship etween the sides nd ngles of n olique tringle You hve investigted the reltionships etween the sides nd ngles of right tringles. Now ou will investigte reltionships etween the sides nd ngles of nonright, or olique, tringles. Investigtion: Olique Tringles Step 1 Drw n ute tringle AC. Lel the side opposite A s, the side opposite s, nd the side opposite C s. Then, drw the ltitude from A to C. Lel the height h. At right is one emple. Step From this digrm, ou n write the following equtions: A h sin h, or h sin C sin C h, or h sin C euse oth sin nd sin C re equl to h, the re equl to eh other. Tht is, sin sin C Dividing oth sides of the eqution ove gives sin sin C Step 3 Now, drw the ltitude from to AC nd lel the height j. Using method similr to tht in Step, ou should find tht sin A sin C (Mke sure ou n derive this eqution on our own!) Steps nd 5 You n omine the proportions from Steps nd 3 to write n etended proportion: sin A sin sin C The tringle ou drew in Step 1 ws ute. Do ou think the sme proportion will e true for otuse tringles? Step 6 Drw n otuse tringle AC nd mesure eh ngle nd side. A At right is one emple. 3 m Find sin A, sin sin C, nd for our tringle. For the tringle t right: sin A sin sin So, it ppers tht sin A sin sin 3 3 sin C 0.13 sin C sin holds for otuse tringles s well m 16 3 C m Disovering Advned Alger Condensed Lessons CHAPTER Kendll Hunt Pulishing (ontinued)

4 Lesson 1. The Lw of Sines (ontinued) Emple A in our ook pplies wht ou lerned in the investigtion to rel-world prolem. Red the emple refull. The reltionship ou disovered in the investigtion is lled the Lw of Sines. It is summrized in the Lw of Sines o in our ook. Emple shows how to ppl the Lw of Sines to find n unknown side length in tringle when ou know the mesures of two ngles nd the length of one side. Red the emple refull. Test our understnding finding the length of side AC. (Hint: You ll need to find the mesure of first.) You should find tht the length of AC is out 15. m. You n lso use the Lw of Sines to find n unknown ngle mesure when ou know two side lengths nd the mesure of the ngle opposite one of the sides. However, in this se ou m find more thn one solution. To help ou understnd wh there m e more thn one solution, look t the digrms on pge 693 of our ook nd red Emple C. Here is nother emple. EXAMPLE Solution In AC, the mesure of A is 30, the length of side A is 8 m, nd the length of side C is 5 m. Sketh nd lel two tringles tht fit this desription. For eh tringle, find the mesures of nd C nd the length of side AC. The two possiilities re shown elow. A 8 m 5 m 8 m A C C 5 m To find one possile mesure for C, use the Lw of Sines. sin 30 5 sin C 8 sin C 8 sin 30 5 C sin 1 8 sin The mesure of C is 53.1, so the mesure of is 180 ( ), or To find the length of AC, use the Lw of Sines gin. sin 30 5 sin sin m sin 30 The length of AC is 9.9 m. The other possile mesure for C is the supplement of 53.1, or The mesure of is then 180 ( ), or 3.1. Use the Lw of Sines to find the length of AC. sin 30 5 sin sin m sin 30 The length of AC is 3.9 m. 178 CHAPTER 1 Disovering Advned Alger Condensed Lessons 010 Kendll Hunt Pulishing

5 CONDENSED LESSON 1.3 The Lw of Cosines In this lesson ou will use the Lw of Cosines to find unknown mesures of tringle when ou know two side lengths nd the mesure of the inluded ngle use the Lw of Cosines to find unknown mesures of tringle when ou know three side lengths You n use the Lw of Sines to find side lengths or ngle mesures of tringle if ou know either two ngle mesures nd one side length or two side lengths nd the mesure of the ngle opposite one of those sides. In Emple A in our ook, ou re given two side lengths nd the mesure of the ngle etween the sides, nd ou must find the length of the third side. The Lw of Sines nnot e pplied in this sitution. Work through the solution to see how to find the unknown side length. If ou use the proedure in Emple A in generl se where ou re given two side lengths, nd, of tringle, AC, nd the mesure of the inluded ngle, C, ou get the Lw of Cosines: os C where is opposite C. Notie tht this looks like the Pthgoren Theorem with n etr term, os C. (In ft, if C is right ngle, then os C is 0 nd the eqution eomes the Pthgoren Theorem.). Red the tet in the Lw of Cosines o on pge 699 in our ook nd stud the digrms fter the o. Investigtion: Around the Corner Red the investigtion in our ook. If ou hve the mterils nd some people to help ou, omplete the investigtion. If not, ou n use the digrm t right. Complete the investigtion on our own, nd then ompre our results to those given. You know the lengths of two sides nd the mesure of n inluded ngle, so ou n use the Lw of Cosines to find the length of the third side. C m 3.5 m A os C The Lw of Cosines..5 (.5)() os 3 Sustitute the known vlues os 3 Multipl os 3 Solve for Evlute. The two towns re out 1.71 meters prt. (ontinued) Disovering Advned Alger Condensed Lessons CHAPTER Kendll Hunt Pulishing

6 Lesson 1.3 The Lw of Cosines (ontinued) To find the unknown mesures in Emple, the Lw of Cosines is pplied twie. Tr to find the unknown mesures ourself, nd then red the solution. In oth the investigtion nd Emple, ou re given two side lengths nd the mesure of the inluded ngle. You n lso use the Lw of Cosines if ou know three side lengths. The emple elow shows ou how. EXAMPLE Find the ngle mesures. 5.1 m 3.5 m C.0 m A Solution Strt using the Lw of Cosines finding the mesure of C. os C The Lw of Cosines (5.1)(.0) os C Sustitute the known vlues os C Multipl os C os C Sutrt from oth sides. Solve for os C. C os Tke the inverse osine of oth sides. C 9.5 Evlute. Now, use the Lw of Sines to find the mesure of. sin C sin sin sin sin.0.0 sin The Lw of Sines. Sustitute the known vlues. Solve for sin. sin 1.0 sin Tke the inverse sine of oth sides Evlute. To find the mesure of A, use the ft tht the sum of the ngle mesures of tringle is 180. A 180 ( ) 13. Red the reminder of the lesson in our ook, whih summrizes wht ou hve lerned in this nd the previous lesson. 180 CHAPTER 1 Disovering Advned Alger Condensed Lessons 010 Kendll Hunt Pulishing

7 CONDENSED LESSON 1. Etending Trigonometr In this lesson ou will etend the definitions of sine, osine, nd tngent to inlude ngles of n mesure find the sine, osine, nd tngent of ngles of rottion use referene ngles to find the sine, osine, nd tngent of relted ngles In Lesson 1.1, the definitions given for sine, osine, nd tngent pplied to ute ngles in right tringles. In this lesson, ou will etend the definitions to ppl to n size ngle. Rememer tht ngles in the oordinte plne re mesured strting from the positive -is nd moving ounterlokwise through Qudrnts I, II, III, nd IV. II III I IV Investigtion: Etending Trigonometri Funtions Red the Proedure Note nd stud the emple shown for Step 1. Then work through the investigtion in our ook. After ou re finished, ompre our nswers to the results elow. Mke sure our lultor is set to degrees. Step 1 The smple nswers use the point (, 0) s the strting point for eh ngle. Your nswers for the oordintes nd the length of the segment will vr depending on the strting point ou hose, ut our results for the sine, osine, nd tngent should mth these results sin , os , nd tn The oordintes of the rotted point re out (.8,.8). The length of the segment is out (.8) units.. 10 sin , os , nd tn The oordintes of the rotted point re out ( 3.5, ). The length of the segment is out ( 3.5) ( ).03 units (ontinued) Disovering Advned Alger Condensed Lessons CHAPTER Kendll Hunt Pulishing

8 Lesson 1. Etending Trigonometr (ontinued). 70 sin 70 1, os 70 0, nd tn 70 is undefined. The oordintes of the rotted point re (0, ). The length of the segment is 0 ( ) units. d. 30 sin , os , nd tn The oordintes of the rotted point re out (3.1,.6). The length of the segment is out 3.1 (.6).05 units. e. 100 sin , os , nd tn The oordintes of the rotted point re out ( 0.7, 3.9). The length of the segment is out ( 0.7) ( 3.9) 3.96 units. Step The results re summrized elow. From these results ou might hpothesize tht sine is -oordinte -oordinte, osine is -oordinte, nd tngent is segment length segment length Angle Sine Cosine Tngent is undefined oordinte. (ontinued) 18 CHAPTER 1 Disovering Advned Alger Condensed Lessons 010 Kendll Hunt Pulishing

9 Lesson 1. Etending Trigonometr (ontinued) Step 3 ( 3, 1) The length of the segment is ( 3) Using the method from Step, 1 sin A, os A 3 10, nd tn A The lultor gives sin nd tn This ngle is in Qudrnt I, so it doesn t mth the digrm. However, using the lultor, os This ngle ppers to mth the digrm. Step The definitions re loted in the definition o on pge 707 of our ook. Red these definitions refull. Red the prgrph efore Emple A, nd then work through Emples A nd in our ook. If ou need to review speil right tringles, red Refreshing Your Skills for Chpter 1 in our ook. elow is nother emple similr to Emple A. EXAMPLE Solution Find the sine, osine, nd tngent of 150 without lultor. Rotte point ounterlokwise 150 from the positive -is. The imge of the point is in Qudrnt II, 30 ove the -is. The referene ngle is 30. The sine, osine, nd tngent of 30 referene ngle re, respetivel, 1_, 3, nd euse the -oordinte is negtive nd the -oordinte is positive in Qudrnt II, sin , os 150, nd tn Disovering Advned Alger Condensed Lessons CHAPTER Kendll Hunt Pulishing

10

11 CONDENSED LESSON 1.5 Introdution to Vetors In this lesson ou will understnd vetors s direted distnes represent ddition, sutrtion, nd slr multiplition of vetors use vetors to solve prolems onvert vetors from one form to nother Some quntities, suh s distne, veloit, nd elertion, n hve diretions ssoited with them. These direted quntities n e represented vetors, whih n e thought of s direted line segments. The line segment hs length, lled the mgnitude, nd diretion. You n represent vetors s segment with n rrowhed t one end, lled the hed or tip. The til is the other end of the vetor. Vetors n e represented in severl ws. The polr form of vetor gives the mgnitude nd the ngle the vetor mkes with the positive -is. For emple, represents vetor 3 units long direted 150 ounterlokwise from the positive -is. The retngulr form of vetor gives the horizontl nd 3 3 vertil hnge from the til to the hed. For emple,, 3 represents 3 3 horizontl hnge of nd vertil hnge of _ 3. Equivlent vetors hve the sme mgnitude nd diretion, no mtter where 3 3 the re loted in the oordinte plne nd, 3_ re equivlent vetors. The investigtion eplores some of the properties of vetor ddition nd sutrtion. Note tht _ nd re two ws to designte vetor. In the eqution, the oldfe letters,, nd represent vetors, nd is the resultnt vetor of the lultion. Investigtion: Vetor Addition nd Sutrtion Work through the whole investigtion in our ook, nd then ompre our results to those elow. Steps The retngulr form of is 6,. Step i. 6 ii. 5 d 3 e The retngulr form of is 6,. The retngulr form of is 0, 1. (ontinued) Disovering Advned Alger Condensed Lessons CHAPTER Kendll Hunt Pulishing

12 Lesson 1.5 Introdution to Vetors (ontinued) iii. 3 iv. 6 f e 6 The retngulr form of is 3, 1. The retngulr form of is 5,. Step 5 If 1, nd 1,, then the sum is 1, 1, 1 1,. Step 6 i ii iii. e 3 d e d 5 iv. 3 e f e f 3 3 Step 7 If 1, nd 1,, then the differene is 1, 1, 1 1,. Step 8 If 1, nd k is slr, then the produt k is k 1, k 1, k. Step 9 The mgnitudes of nd re 3 13 nd If 1,, then the mgnitude of, denoted, is 1. Vetors re useful for representing motion. Red Emple A to eplore n pplition of vetor ddition. Sometimes the polr form of vetor is more pproprite. Emple eplins how to onvert from retngulr form to polr form. Red Emple nd mke sure ou understnd how to onvert from retngulr to polr form. Red the tet following Emple. e sure ou understnd how to hnge ering to n ngle tht gives diretion of vetor in polr form. In Emple C, the vetors must e onverted from polr form to retngulr form to dd them. Work refull through Emple C. 186 CHAPTER 1 Disovering Advned Alger Condensed Lessons 010 Kendll Hunt Pulishing

13 CONDENSED LESSON 1.6 Prmetri Equtions In this lesson ou will use prmeter to write prmetri equtions tht seprtel define nd grph prmetri equtions use prmetri equtions to model rel-world prolems So fr, ou hve used equtions to relte nd to eh other. Sometimes ou wnt to epress nd s seprte funtions of third vrile, t, lled the prmeter. These prmetri equtions provide ou with more informtion nd etter ontrol over wht points ou plot. You n use prmetri equtions to epress - nd -oordintes s funtions of time. Emple A in our ook shows how to use prmetri equtions to model motion prolem. Red Emple A nd its solution refull. Then red the following emple. EXAMPLE A Solution Jmes is rowing ot 30 ft ross river. He rows t rte of 1 ft/s diretl towrd the opposite shore. The urrent moves perpendiulr to his diretion of rowing t rte of 3 ft/s. The post where Jmes wnts to tie up his rowot is 100 ft downstrem from his strting point. Will Jmes mke it to the other side of the river efore he psses the post? Let represent the distne in feet the ot moves due to the urrent, let represent the distne in feet Jmes hs rowed ross the river, nd let t represent the time in seonds. Then 3t nd t. Grph this pir of equtions on our lultor. See Clultor Note 1C to lern how to enter nd grph prmetri equtions. Use n pproprite window for the ontet. You n piture the post t the point (100, 30). If ou tre point on the grph, ou will see tht Jmes will hve 10 feet to spre efore he rehes the post. Prmetri equtions n help ou model omplited situtions involving motion. Mn pirs of prmetri equtions n e written s single eqution using onl nd. If ou rewrite prmetri model s single eqution, then ou ll hve two different ws to stud sitution. (ontinued) Disovering Advned Alger Condensed Lessons CHAPTER Kendll Hunt Pulishing

14 Lesson 1.6 Prmetri Equtions (ontinued) Investigtion: Prmetri Wlk Steps 1 nd Red Steps 1 nd nd the Proedure Note of the investigtion in our ook. Mke sure ou n visulize wht is going on: A segment is mrked on oordinte grid. As person wlks long the segment, one motion sensor (held reorder X) is reording how the -oordinte of the person s pth hnges nd one sensor (held reorder Y) is reord ing how the -oordinte of the person s pth hnges. Enter the smple dt in our lultor nd omplete the rest of the investigtion on our own. Then ompre our results to those elow. Step 3 Use our lultor to find the medinmedin lines. The medin-medin line for the (t, ) dt is ˆ 0.18t 1.8. Dt olleted reorder X t Dt olleted reorder Y t Step The medin-medin line for the (t, ) dt is ŷ 0.10t Step 5 The grph t right shows plot of the (, ) vlues, long with grphs of the prmetri funtions 0.18t 1.8 nd 0.10t The prmetri funtions seem to fit the dt. 1.8 Step 6 Solving ˆ 0.18t 1.8 for t gives t Sustitute this epression for t into the eqution for : ŷ Step 7 The grph t right shows the (, ) dt nd the funtion ŷ from Step 6. Step 8 Eliminting the prmeter gives the sme grph, ut ou lose the informtion out the time, nd ou nnot limit the vlues of t to show onl the segment on the line tht ws tull wlked. (ontinued) 188 CHAPTER 1 Disovering Advned Alger Condensed Lessons 010 Kendll Hunt Pulishing

15 Lesson 1.6 Prmetri Equtions (ontinued) Red the tet following the investigtion nd Emple. Emple eplins how to model projetile motion prmetrill. The emple tht follows lso onerns projetile motion. EXAMPLE Peter punts footll t n ngle of 55 so tht it hs n initil veloit of 75 ft/s. If his foot ontts the ll t height 3.5 ft ove the ground, how fr does the ll trvel horizontll efore it hits the ground? Solution Drw piture nd find the - nd -omponents of the initil veloit. os sin os sin 55 The horizontl motion is ffeted onl the initil speed nd ngle, so the horizontl distne is modeled 75t os 55. The vertil motion is ffeted the fore of grvit nd the initil height. Its eqution is 16t 75t sin To find when the ll hits the ground, find t when is 0. 16t 75t sin t 75 sin 55 (75 sin 55 ) ( 16)(3.5) ( 16) t or t ft/s 55 Onl the positive nswer mkes sense in this sitution. The ll hits the ground out seonds fter it is kiked. To find how fr the ll hs trveled, sustitute this t-vlue into the eqution for : 75(3.896) os The ll trvels out ft, or 56 d, horizontll. Disovering Advned Alger Condensed Lessons CHAPTER Kendll Hunt Pulishing

16

The remaining two sides of the right triangle are called the legs of the right triangle.

The remaining two sides of the right triangle are called the legs of the right triangle. 10 MODULE 6. RADICAL EXPRESSIONS 6 Pythgoren Theorem The Pythgoren Theorem An ngle tht mesures 90 degrees is lled right ngle. If one of the ngles of tringle is right ngle, then the tringle is lled right

More information

SECTION 7-2 Law of Cosines

SECTION 7-2 Law of Cosines 516 7 Additionl Topis in Trigonometry h d sin s () tn h h d 50. Surveying. The lyout in the figure t right is used to determine n inessile height h when seline d in plne perpendiulr to h n e estlished

More information

How to Graphically Interpret the Complex Roots of a Quadratic Equation

How to Graphically Interpret the Complex Roots of a Quadratic Equation Universit of Nersk - Linoln DigitlCommons@Universit of Nersk - Linoln MAT Em Epositor Ppers Mth in the Middle Institute Prtnership 7-007 How to Grphill Interpret the Comple Roots of Qudrti Eqution Crmen

More information

1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5.

1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5. . Definition, Bsi onepts, Types. Addition nd Sutrtion of Mtries. Slr Multiplition. Assignment nd nswer key. Mtrix Multiplition. Assignment nd nswer key. Determinnt x x (digonl, minors, properties) summry

More information

PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * challenge questions

PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * challenge questions PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * hllenge questions e The ll will strike the ground 1.0 s fter it is struk. Then v x = 20 m s 1 nd v y = 0 + (9.8 m s 2 )(1.0 s) = 9.8 m s 1 The speed

More information

Lesson 2.1 Inductive Reasoning

Lesson 2.1 Inductive Reasoning Lesson.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 1, 16,,. 400, 00, 100, 0,,,. 1 8, 7, 1, 4,, 4.,,, 1, 1, 0,,. 60, 180, 10,

More information

State the size of angle x. Sometimes the fact that the angle sum of a triangle is 180 and other angle facts are needed. b y 127

State the size of angle x. Sometimes the fact that the angle sum of a triangle is 180 and other angle facts are needed. b y 127 ngles 2 CHTER 2.1 Tringles Drw tringle on pper nd lel its ngles, nd. Ter off its orners. Fit ngles, nd together. They mke stright line. This shows tht the ngles in this tringle dd up to 180 ut it is not

More information

Practice Test 2. a. 12 kn b. 17 kn c. 13 kn d. 5.0 kn e. 49 kn

Practice Test 2. a. 12 kn b. 17 kn c. 13 kn d. 5.0 kn e. 49 kn Prtie Test 2 1. A highwy urve hs rdius of 0.14 km nd is unnked. A r weighing 12 kn goes round the urve t speed of 24 m/s without slipping. Wht is the mgnitude of the horizontl fore of the rod on the r?

More information

Sine and Cosine Ratios. For each triangle, find (a) the length of the leg opposite lb and (b) the length of the leg adjacent to lb.

Sine and Cosine Ratios. For each triangle, find (a) the length of the leg opposite lb and (b) the length of the leg adjacent to lb. - Wht You ll ern o use sine nd osine to determine side lengths in tringles... nd Wh o use the sine rtio to estimte stronomil distnes indiretl, s in Emple Sine nd osine tios hek Skills You ll Need for Help

More information

1. Area under a curve region bounded by the given function, vertical lines and the x axis.

1. Area under a curve region bounded by the given function, vertical lines and the x axis. Ares y Integrtion. Are uner urve region oune y the given funtion, vertil lines n the is.. Are uner urve region oune y the given funtion, horizontl lines n the y is.. Are etween urves efine y two given

More information

Geometry 7-1 Geometric Mean and the Pythagorean Theorem

Geometry 7-1 Geometric Mean and the Pythagorean Theorem Geometry 7-1 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the

More information

Angles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example

Angles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example 2.1 Angles Reognise lternte n orresponing ngles Key wors prllel lternte orresponing vertilly opposite Rememer, prllel lines re stright lines whih never meet or ross. The rrows show tht the lines re prllel

More information

Words Symbols Diagram. abcde. a + b + c + d + e

Words Symbols Diagram. abcde. a + b + c + d + e Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To

More information

Section 5-5 Solving Right Triangles*

Section 5-5 Solving Right Triangles* 5-5 Solving Right Tringles 379 79. Geometry. The re of retngulr n-sided polygon irumsried out irle of rdius is given y A n tn 80 n (A) Find A for n 8, n 00, n,000, nd n 0,000. Compute eh to five deiml

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

Radius of the Earth - Radii Used in Geodesy James R. Clynch Naval Postgraduate School, 2002

Radius of the Earth - Radii Used in Geodesy James R. Clynch Naval Postgraduate School, 2002 dius of the Erth - dii Used in Geodesy Jmes. Clynh vl Postgrdute Shool, 00 I. Three dii of Erth nd Their Use There re three rdii tht ome into use in geodesy. These re funtion of ltitude in the ellipsoidl

More information

Maximum area of polygon

Maximum area of polygon Mimum re of polygon Suppose I give you n stiks. They might e of ifferent lengths, or the sme length, or some the sme s others, et. Now there re lots of polygons you n form with those stiks. Your jo is

More information

Right-angled triangles

Right-angled triangles 13 13A Pythgors theorem 13B Clulting trigonometri rtios 13C Finding n unknown side 13D Finding ngles 13E Angles of elevtion nd depression Right-ngled tringles Syllus referene Mesurement 4 Right-ngled tringles

More information

Quick Guide to Lisp Implementation

Quick Guide to Lisp Implementation isp Implementtion Hndout Pge 1 o 10 Quik Guide to isp Implementtion Representtion o si dt strutures isp dt strutures re lled S-epressions. The representtion o n S-epression n e roken into two piees, the

More information

Calculating Principal Strains using a Rectangular Strain Gage Rosette

Calculating Principal Strains using a Rectangular Strain Gage Rosette Clulting Prinipl Strins using Retngulr Strin Gge Rosette Strin gge rosettes re used often in engineering prtie to determine strin sttes t speifi points on struture. Figure illustrtes three ommonly used

More information

Ratio and Proportion

Ratio and Proportion Rtio nd Proportion Rtio: The onept of rtio ours frequently nd in wide vriety of wys For exmple: A newspper reports tht the rtio of Repulins to Demorts on ertin Congressionl ommittee is 3 to The student/fulty

More information

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324 A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................

More information

Vectors Summary. Projection vector AC = ( Shortest distance from B to line A C D [OR = where m1. and m

Vectors Summary. Projection vector AC = ( Shortest distance from B to line A C D [OR = where m1. and m . Slr prout (ot prout): = osθ Vetors Summry Lws of ot prout: (i) = (ii) ( ) = = (iii) = (ngle etween two ientil vetors is egrees) (iv) = n re perpeniulr Applitions: (i) Projetion vetor: B Length of projetion

More information

Experiment 6: Friction

Experiment 6: Friction Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions. Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

More information

MATH PLACEMENT REVIEW GUIDE

MATH PLACEMENT REVIEW GUIDE MATH PLACEMENT REVIEW GUIDE This guie is intene s fous for your review efore tking the plement test. The questions presente here my not e on the plement test. Although si skills lultor is provie for your

More information

EXAMPLE EXAMPLE. Quick Check EXAMPLE EXAMPLE. Quick Check. EXAMPLE Real-World Connection EXAMPLE

EXAMPLE EXAMPLE. Quick Check EXAMPLE EXAMPLE. Quick Check. EXAMPLE Real-World Connection EXAMPLE - Wht You ll Lern To use the Pthgoren Theorem To use the onverse of the Pthgoren Theorem... nd Wh To find the distne etween two doks on lke, s in Emple The Pthgoren Theorem nd Its onverse hek Skills You

More information

SOLVING EQUATIONS BY FACTORING

SOLVING EQUATIONS BY FACTORING 316 (5-60) Chpter 5 Exponents nd Polynomils 5.9 SOLVING EQUATIONS BY FACTORING In this setion The Zero Ftor Property Applitions helpful hint Note tht the zero ftor property is our seond exmple of getting

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting

More information

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right.

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right. Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction

More information

The Pythagorean Theorem

The Pythagorean Theorem The Pythgoren Theorem Pythgors ws Greek mthemtiin nd philosopher, orn on the islnd of Smos (. 58 BC). He founded numer of shools, one in prtiulr in town in southern Itly lled Crotone, whose memers eventully

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define

More information

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a. Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles

More information

Pure C4. Revision Notes

Pure C4. Revision Notes Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd

More information

EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

More information

Angles and Triangles

Angles and Triangles nges nd Tringes n nge is formed when two rys hve ommon strting point or vertex. The mesure of n nge is given in degrees, with ompete revoution representing 360 degrees. Some fmiir nges inude nother fmiir

More information

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

Tallahassee Community College. Simplifying Radicals

Tallahassee Community College. Simplifying Radicals Tllhssee Communit College Simplifing Rdils The squre root of n positive numer is the numer tht n e squred to get the numer whose squre root we re seeking. For emple, 1 euse if we squre we get 1, whih is

More information

50 MATHCOUNTS LECTURES (10) RATIOS, RATES, AND PROPORTIONS

50 MATHCOUNTS LECTURES (10) RATIOS, RATES, AND PROPORTIONS 0 MATHCOUNTS LECTURES (0) RATIOS, RATES, AND PROPORTIONS BASIC KNOWLEDGE () RATIOS: Rtios re use to ompre two or more numers For n two numers n ( 0), the rtio is written s : = / Emple : If 4 stuents in

More information

Or more simply put, when adding or subtracting quantities, their uncertainties add.

Or more simply put, when adding or subtracting quantities, their uncertainties add. Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re

More information

Module 5. Three-phase AC Circuits. Version 2 EE IIT, Kharagpur

Module 5. Three-phase AC Circuits. Version 2 EE IIT, Kharagpur Module 5 Three-hse A iruits Version EE IIT, Khrgur esson 8 Three-hse Blned Suly Version EE IIT, Khrgur In the module, ontining six lessons (-7), the study of iruits, onsisting of the liner elements resistne,

More information

Lecture 15 - Curve Fitting Techniques

Lecture 15 - Curve Fitting Techniques Lecture 15 - Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting - motivtion For root finding, we used given function to identify where it crossed zero where does fx

More information

End of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below.

End of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below. End of term: TEST A You will need penil nd ruler. Yer Nme Clss Dte Complete the missing numers in the sequenes elow. 8 30 3 28 2 9 25 00 75 25 2 Put irle round ll of the following shpes whih hve 3 shded.

More information

Volumes by Cylindrical Shells: the Shell Method

Volumes by Cylindrical Shells: the Shell Method olumes Clinril Shells: the Shell Metho Another metho of fin the volumes of solis of revolution is the shell metho. It n usull fin volumes tht re otherwise iffiult to evlute using the Dis / Wsher metho.

More information

Square Roots Teacher Notes

Square Roots Teacher Notes Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this

More information

MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!

MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent! MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more

More information

Lesson 4.1 Triangle Sum Conjecture

Lesson 4.1 Triangle Sum Conjecture Lesson 4.1 ringle um onjecture Nme eriod te n ercises 1 9, determine the ngle mesures. 1. p, q 2., y 3., b 31 82 p 98 q 28 53 y 17 79 23 50 b 4. r, s, 5., y 6. y t t s r 100 85 100 y 30 4 7 y 31 7. s 8.

More information

Lesson 12.1 Trigonometric Ratios

Lesson 12.1 Trigonometric Ratios Lesson 12.1 rigonometric Rtios Nme eriod Dte In Eercises 1 6, give ech nswer s frction in terms of p, q, nd r. 1. sin 2. cos 3. tn 4. sin Q 5. cos Q 6. tn Q p In Eercises 7 12, give ech nswer s deciml

More information

GRADE 4. Fractions WORKSHEETS

GRADE 4. Fractions WORKSHEETS GRADE Frtions WORKSHEETS Types of frtions equivlent frtions This frtion wll shows frtions tht re equivlent. Equivlent frtions re frtions tht re the sme mount. How mny equivlent frtions n you fin? Lel eh

More information

1 Fractions from an advanced point of view

1 Fractions from an advanced point of view 1 Frtions from n vne point of view We re going to stuy frtions from the viewpoint of moern lger, or strt lger. Our gol is to evelop eeper unerstning of wht n men. One onsequene of our eeper unerstning

More information

LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.

LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes. LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 64-83.

More information

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.

More information

4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A

4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A Geometry: Shpes. Circumference nd re of circle HOMEWORK D C 3 5 6 7 8 9 0 3 U Find the circumference of ech of the following circles, round off your nswers to dp. Dimeter 3 cm Rdius c Rdius 8 m d Dimeter

More information

AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.

More information

Unit 6: Exponents and Radicals

Unit 6: Exponents and Radicals Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -

More information

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1. Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose

More information

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1 PROBLEMS - APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.

More information

The Math Learning Center PO Box 12929, Salem, Oregon 97309 0929 Math Learning Center

The Math Learning Center PO Box 12929, Salem, Oregon 97309 0929  Math Learning Center Resource Overview Quntile Mesure: Skill or Concept: 1010Q Determine perimeter using concrete models, nonstndrd units, nd stndrd units. (QT M 146) Use models to develop formuls for finding res of tringles,

More information

Chapter. Contents: A Constructing decimal numbers

Chapter. Contents: A Constructing decimal numbers Chpter 9 Deimls Contents: A Construting deiml numers B Representing deiml numers C Deiml urreny D Using numer line E Ordering deimls F Rounding deiml numers G Converting deimls to frtions H Converting

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

Heron s Formula for Triangular Area

Heron s Formula for Triangular Area Heron s Formul for Tringulr Are y Christy Willims, Crystl Holom, nd Kyl Gifford Heron of Alexndri Physiist, mthemtiin, nd engineer Tught t the museum in Alexndri Interests were more prtil (mehnis, engineering,

More information

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn 33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

More information

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the

More information

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001 CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic

More information

Cypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period:

Cypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period: Nme: SOLUTIONS Dte: Period: Directions: Solve ny 5 problems. You my ttempt dditionl problems for extr credit. 1. Two blocks re sliding to the right cross horizontl surfce, s the drwing shows. In Cse A

More information

SOLVING QUADRATIC EQUATIONS BY FACTORING

SOLVING QUADRATIC EQUATIONS BY FACTORING 6.6 Solving Qudrti Equtions y Ftoring (6 31) 307 In this setion The Zero Ftor Property Applitions 6.6 SOLVING QUADRATIC EQUATIONS BY FACTORING The tehniques of ftoring n e used to solve equtions involving

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one. 5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics. W02D3_0 Group Problem: Pulleys and Ropes Constraint Conditions

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics. W02D3_0 Group Problem: Pulleys and Ropes Constraint Conditions MSSCHUSES INSIUE OF ECHNOLOGY Deprtment of hysics 8.0 W02D3_0 Group roblem: ulleys nd Ropes Constrint Conditions Consider the rrngement of pulleys nd blocks shown in the figure. he pulleys re ssumed mssless

More information

Version 001 CIRCUITS holland (1290) 1

Version 001 CIRCUITS holland (1290) 1 Version CRCUTS hollnd (9) This print-out should hve questions Multiple-choice questions my continue on the next column or pge find ll choices efore nswering AP M 99 MC points The power dissipted in wire

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100 hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

Binary Representation of Numbers Autar Kaw

Binary Representation of Numbers Autar Kaw Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

Trigonometry & Pythagoras Theorem

Trigonometry & Pythagoras Theorem Trigonometry & Pythagoras Theorem Mathematis Skills Guide This is one of a series of guides designed to help you inrease your onfidene in handling Mathematis. This guide ontains oth theory and exerises

More information

c b 5.00 10 5 N/m 2 (0.120 m 3 0.200 m 3 ), = 4.00 10 4 J. W total = W a b + W b c 2.00

c b 5.00 10 5 N/m 2 (0.120 m 3 0.200 m 3 ), = 4.00 10 4 J. W total = W a b + W b c 2.00 Chter 19, exmle rolems: (19.06) A gs undergoes two roesses. First: onstnt volume @ 0.200 m 3, isohori. Pressure inreses from 2.00 10 5 P to 5.00 10 5 P. Seond: Constnt ressure @ 5.00 10 5 P, isori. olume

More information

Clause Trees: a Tool for Understanding and Implementing Resolution in Automated Reasoning

Clause Trees: a Tool for Understanding and Implementing Resolution in Automated Reasoning Cluse Trees: Tool for Understnding nd Implementing Resolution in Automted Resoning J. D. Horton nd Brue Spener University of New Brunswik, Frederiton, New Brunswik, Cnd E3B 5A3 emil : jdh@un. nd spener@un.

More information

Review guide for the final exam in Math 233

Review guide for the final exam in Math 233 Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered

More information

Lecture 3: orientation. Computer Animation

Lecture 3: orientation. Computer Animation Leture 3: orienttion Computer Animtion Mop tutoril sessions Next Thursdy (Feb ) Tem distribution: : - :3 - Tems 7, 8, 9 :3 - : - Tems nd : - :3 - Tems 5 nd 6 :3 - : - Tems 3 nd 4 Pper ssignments Pper ssignment

More information

Seeking Equilibrium: Demand and Supply

Seeking Equilibrium: Demand and Supply SECTION 1 Seeking Equilirium: Demnd nd Supply OBJECTIVES KEY TERMS TAKING NOTES In Setion 1, you will explore mrket equilirium nd see how it is rehed explin how demnd nd supply intert to determine equilirium

More information

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2 7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6

More information

Enterprise Digital Signage Create a New Sign

Enterprise Digital Signage Create a New Sign Enterprise Digitl Signge Crete New Sign Intended Audiene: Content dministrtors of Enterprise Digitl Signge inluding stff with remote ess to sign.pitt.edu nd the Content Mnger softwre pplition for their

More information

5.6 POSITIVE INTEGRAL EXPONENTS

5.6 POSITIVE INTEGRAL EXPONENTS 54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section

More information

LISTENING COMPREHENSION

LISTENING COMPREHENSION PORG, přijímí zkoušky 2015 Angličtin B Reg. číslo: Inluded prts: Points (per prt) Points (totl) 1) Listening omprehension 2) Reding 3) Use of English 4) Writing 1 5) Writing 2 There re no extr nswersheets

More information

If two triangles are perspective from a point, then they are also perspective from a line.

If two triangles are perspective from a point, then they are also perspective from a line. Mth 487 hter 4 Prtie Prolem Solutions 1. Give the definition of eh of the following terms: () omlete qudrngle omlete qudrngle is set of four oints, no three of whih re olliner, nd the six lines inident

More information

The invention of line integrals is motivated by solving problems in fluid flow, forces, electricity and magnetism.

The invention of line integrals is motivated by solving problems in fluid flow, forces, electricity and magnetism. Instrutor: Longfei Li Mth 43 Leture Notes 16. Line Integrls The invention of line integrls is motivted by solving problems in fluid flow, fores, eletriity nd mgnetism. Line Integrls of Funtion We n integrte

More information

Warm-up for Differential Calculus

Warm-up for Differential Calculus Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:

More information

Released Assessment Questions, 2015 QUESTIONS

Released Assessment Questions, 2015 QUESTIONS Relesed Assessmet Questios, 15 QUESTIONS Grde 9 Assessmet of Mthemtis Ademi Red the istrutios elow. Alog with this ooklet, mke sure you hve the Aswer Booklet d the Formul Sheet. You my use y spe i this

More information

CHAPTER 31 CAPACITOR

CHAPTER 31 CAPACITOR . Given tht Numer of eletron HPTER PITOR Net hrge Q.6 9.6 7 The net potentil ifferene L..6 pitne v 7.6 8 F.. r 5 m. m 8.854 5.4 6.95 5 F... Let the rius of the is R re R D mm m 8.85 r r 8.85 4. 5 m.5 m

More information

Vectors 2. 1. Recap of vectors

Vectors 2. 1. Recap of vectors Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

More information

NCERT INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS. Trigonometric Ratios of the angle A in a triangle ABC right angled at B are defined as:

NCERT INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS. Trigonometric Ratios of the angle A in a triangle ABC right angled at B are defined as: INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS (A) Min Concepts nd Results Trigonometric Rtios of the ngle A in tringle ABC right ngled t B re defined s: side opposite to A BC sine of A = sin A = hypotenuse

More information

Double Integrals over General Regions

Double Integrals over General Regions Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing

More information

Section 7-4 Translation of Axes

Section 7-4 Translation of Axes 62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

ONLINE PAGE PROOFS. Trigonometry. 6.1 Overview. topic 6. Why learn this? What do you know? Learning sequence. measurement and geometry

ONLINE PAGE PROOFS. Trigonometry. 6.1 Overview. topic 6. Why learn this? What do you know? Learning sequence. measurement and geometry mesurement nd geometry topic 6 Trigonometry 6.1 Overview Why lern this? Pythgors ws gret mthemticin nd philosopher who lived in the 6th century BCE. He is est known for the theorem tht ers his nme. It

More information

15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style

15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style The men vlue nd the root-men-squre vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time

More information

Variable Dry Run (for Python)

Variable Dry Run (for Python) Vrile Dr Run (for Pthon) Age group: Ailities ssumed: Time: Size of group: Focus Vriles Assignment Sequencing Progrmming 7 dult Ver simple progrmming, sic understnding of ssignment nd vriles 20-50 minutes

More information