2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES"

Transcription

1 . TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an angle α is said to be in standad position if its vete is at the oigin O and its initial side coincides with the positive ais (Figue.). An angle is said to be in a cetain quadant if, when the angle is in standad position, the teminal side lies in that quadant. Fo instance, a 6 angle lies in quadant I o simpl that it is a quadant I angle. As Figue b shows, an angle of 8 is a quadant II angle. If the teminal side of an angle in standad position lies along eithe the ais o the ais, then the angle is called quadantal. Fo eample, 60, 0, 80, 90, 0, 90, 80, 0, 60 ae quadantal angles. Evidentl, an angle is quadantal if and onl π if its measue is an intege multiple of 90 ( o adians). Figue. α is in standad position teminal side α initial side Figue. (a) quadant I angle (b) quadant II angle teminal side 6 teminal side 8 9

2 Definition.: Tigonometic Functions of a Geneal Angle Let θ be an angle in standad position and suppose that (, ) is an point othe than ( 0, 0 ) on the teminal side of θ (Figue.). If + is the distance between (, ) and ( 0, 0 ), then the si tigonometic functions of θ ae defined b Figue. sin θ cos θ tan θ csc θ sec θ cot θ.(, ) θ povided that the denominatos ae not zeo. O Using simila tiangles, ou can see that the values of the si tigonometic functions in Definition. depend onl on the angle θ and not on the choice of the point (, ) on the teminal side of θ. Eample Evaluate the si tigonometic functions of the angle θ in standad position if the teminal side of θ contains the point (, ) (, -). Hee,, -, and Thus, + + ( ). sin θ cos θ tan θ csc θ sec θ cot θ. You can detemine the algebaic signs of the tigonometic functions fo angles in the vaious quadants b ecalling the algebaic signs of and in these quadants and 0

3 emembeing that is alwas positive. Fo instance, as Figue. shows, is positive in quadants I and II (whee both and ae positive), and it is negative in quadants III and IV (whee is negative and is positive). B poceeding in a simila wa, ou can detemine the signs of the emaining tigonometic functions in the vaious quadants and thus confim the esults in Table.. Figue. > 0 > 0 > 0 Quadant II > 0 Quadant I < 0 O < 0 < 0 Quadant III < 0 Quadant IV Table. Quadant Containing θ I II III IV Positive Functions All, cscθ tanθ, cotθ, secθ Negative Functions None, secθ, tanθ, cotθ, cscθ,, secθ, cscθ, tanθ, cotθ Eample Find the quadant in which θ lies if tanθ > 0 and < 0. This eample can be woked b using Table.; howeve, athe than eling on the table, we pefe to eason as follows: Let (, ) be a point othe than the oigin on the teminal side of θ (in standad position). Because tanθ > 0, we see that and have the same algebaic sign. Futhemoe, since < 0, it follows that < 0. Because < 0 and < 0, the angle is in quadant III.

4 Recipocal Identities If θ is an angle fo which the functions ae defined, then: (i) cscθ (ii) secθ (iii) cotθ tanθ. Quotient Identities If θ is an angle fo which the functions ae defined, then: tanθ and cotθ. Eample If and of θ. tanθ secθ cotθ cscθ tanθ, find the values of the othe fou tigonometic functions. B using the ecipocal and quotient identities, ou can quickl ecall the algebaic signs of the secant, cosecant, tangent, and cotangent in the fou quadants (Table ), if ou know the algebaic signs of the sine and cosine in these quadants. Anothe impotant identit is deived as follows: Again suppose that θ is an angle in standad position and that (, ) is a point othe than the oigin on the teminal side of θ (Figue 9). Because +, we have +, so (cos θ ) + ( ) + + The elationship: ( ) + ( ) is called the fundamental Pthagoean identit because its deivation involves the fact that +, which is a consequence of the Pthagoean theoem..

5 The fundamental Pthagoean identit is used quite often, and it would be bothesome to wite the paentheses each time fo ( ) and ( ) ; et, if the paentheses wee simpl omitted, the esulting epessions would be misundestood. (Fo instance, is usuall undestood to mean the cosine of the squae of θ.) Theefoe, it is customa to wite cos θ and sin θ to mean ( ) and ( ). Simila notation is used fo the emaining tigonometic functions and fo powes othe than. Thus, cot θ means ( cot θ ), n n sec θ means ( sec θ ), and so foth. With this notation, the fundamental Pthagoean identit becomes cos θ + sin θ. Actuall, thee ae thee Pthagoean identities the fundamental identit and two othes deived fom it. Pthagoean Identities If θ is an angle fo which the functions ae defined, then: (i) θ cos + sin θ (ii) + tan θ sec θ (iii) + cot θ csc θ We alead poved (i). To pove (ii), we divide both sides of (i) b sin θ + cos θ cos θ o +, povided that cos θ 0. Since tanθ and secθ, cos θ to obtain we have that + tan θ θ sec. Identit (iii) is poved b dividing both sides of (i) b sin θ. Eample The value of one of the tigonometic functions of an angle θ is given along with the infomation about the quadant in which θ lies, Find the values of the othe five tigonometic functions of θ :

6 ( a ), θ in quadant II. B the fundamental Pthagoean identit, cos θ + sin θ, so cos θ θ sin Theefoe, cos θ ± ±. 69 Because θ is in quadant II, we know that is negative; hence, -. It follows that tanθ secθ cotθ tanθ cscθ ( b ) tanθ and < 0. Because tanθ < 0 onl in quadants II and IV, and < 0 onl in quadants III and IV, it follows that θ must be in quadant IV. B pat (ii) θ sec + tan θ, so sec θ ± + tan θ ± + (- 8 ) ± Since θ is in quadant IV, secθ > 0; hence, secθ. Because secθ it follows that secθ, θ θ sin Now, tanθ cos 8 so (tanθ )( ) Finall, cscθ and cotθ tanθ ± 9 8. ±.

7 In the applications of tigonomet, and especiall in calculus, it is often necessa to make tigonometic calculations, as we have done in this section, without the use of calculatos o tables. Section Poblems In poblems to 0, sketch two coteminal angles α and β in standad position whose teminal side contains the given point. Aange it so that α is positive, β is negative, and neithe angle eceeds one evolution. In each case, name the quadant in which the angle lies, o indicate that the angle is quadantal.. (, ). (, ), ). (, 0 ). (. (, ) 6. ( 0, ). (, ) 8. (, 0 ) 9. (, ) 0. ( 0, ) In poblems to 8, specif and sketch thee angles that ae coteminal with the given angle in standad position π π.. π π In poblems 6 to 8, evaluate the si tigonometic functions of the angle θ in standad position if the teminal side of θ contains the given point (, ). [Do not use a calculato leave all answes in the fom of a faction o an intege.] In each case, sketch one of the coteminal angles θ. 9. (, ) 0. (, ). (, ). (, ). (, ). (, ). (, ) 6. (, ). (, ) 8. ( 0, ) 9. Is thee an angle θ fo which? Eplain.

8 0. Using simila tiangles, show that the values of the si tigonometic functions in Definition. depend onl on the angle θ and not on the choice of the point (, ) on the teminal side of θ.. In each case, assume that θ is an angle in standad position and find the quadant in which it lies. (a) tanθ > 0 and secθ > 0 (b) > 0 and secθ < 0 (c) > 0 and < 0 (d) secθ > 0 and tanθ < 0 (e) tanθ > 0 and cscθ < 0 (f) < 0 and cscθ < 0 (g) secθ > 0 and cotθ < 0 (h) cotθ > 0 and > 0. Is thee an angle θ fo which > 0 and cscθ < 0? Eplain.. Give the algebaic sign of each of the following. (a) cos 6 (b) sin o π (d) tan (c) sec ( ) (e) cot (g) sec 8π 8π (f) csc π. If θ is an angle fo which the functions ae defined, show that secθ ( )( tanθ ).. If and cos θ, use the ecipocal and quotient identities to find (a) secθ (b) cscθ (c) tanθ (d) cotθ. 6

9 6. If secθ and cscθ, use the ecipocal and quotient identities to find (a) (b) (c) tanθ (d) cotθ. In Poblems to 8, the values of one of the tigonometic functions of an angle θ is given along with infomation about the quadant (Q) in which θ lies. Find the values of the othe five tigonometic functions of θ.., θ in Q I 8., θ in Q IV 9., θ in Q III 0., θ not in Q I., < 0., θ not in Q I. cscθ, θ in Q I. secθ, θ in Q III. tanθ, θ in Q I 6. tanθ, < 0. cotθ, cscθ > 0 8. cscθ, secθ < 0

UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

More information

Trigonometric Functions of Any Angle

Trigonometric Functions of Any Angle Tigonomet Module T2 Tigonometic Functions of An Angle Copight This publication The Nothen Albeta Institute of Technolog 2002. All Rights Reseved. LAST REVISED Decembe, 2008 Tigonometic Functions of An

More information

Trigonometry in the Cartesian Plane

Trigonometry in the Cartesian Plane Tigonomet in the Catesian Plane CHAT Algeba sec. 0. to 0.5 *Tigonomet comes fom the Geek wod meaning measuement of tiangles. It pimail dealt with angles and tiangles as it petained to navigation astonom

More information

4.1 - Trigonometric Functions of Acute Angles

4.1 - Trigonometric Functions of Acute Angles 4.1 - Tigonometic Functions of cute ngles a is a half-line that begins at a point and etends indefinitel in some diection. Two as that shae a common endpoint (o vete) fom an angle. If we designate one

More information

CHAT Pre-Calculus Section 10.7. Polar Coordinates

CHAT Pre-Calculus Section 10.7. Polar Coordinates CHAT Pe-Calculus Pola Coodinates Familia: Repesenting gaphs of equations as collections of points (, ) on the ectangula coodinate sstem, whee and epesent the diected distances fom the coodinate aes to

More information

Skills Needed for Success in Calculus 1

Skills Needed for Success in Calculus 1 Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

More information

2.2. Trigonometric Ratios of Any Angle. Investigate Trigonometric Ratios for Angles Greater Than 90

2.2. Trigonometric Ratios of Any Angle. Investigate Trigonometric Ratios for Angles Greater Than 90 . Tigonometic Ratios of An Angle Focus on... detemining the distance fom the oigin to a point (, ) on the teminal am of an angle detemining the value of sin, cos, o tan given an point (, ) on the teminal

More information

Coordinate Systems L. M. Kalnins, March 2009

Coordinate Systems L. M. Kalnins, March 2009 Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

More information

Originally TRIGONOMETRY was that branch of mathematics concerned with solving triangles using trigonometric ratios which were seen as properties of

Originally TRIGONOMETRY was that branch of mathematics concerned with solving triangles using trigonometric ratios which were seen as properties of Oiginall TRIGONOMETRY was that banch of mathematics concened with solving tiangles using tigonometic atios which wee seen as popeties of tiangles athe than of angles. The wod Tigonomet comes fom the Geek

More information

Section 5-3 Angles and Their Measure

Section 5-3 Angles and Their Measure 5 5 TRIGONOMETRIC FUNCTIONS Section 5- Angles and Thei Measue Angles Degees and Radian Measue Fom Degees to Radians and Vice Vesa In this section, we intoduce the idea of angle and two measues of angles,

More information

Trigonometry Review Workshop 1

Trigonometry Review Workshop 1 Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions

More information

LINES AND TANGENTS IN POLAR COORDINATES

LINES AND TANGENTS IN POLAR COORDINATES LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Pola-coodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and

More information

4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to

4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to . Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate

More information

TRIGONOMETRY REVIEW. The Cosines and Sines of the Standard Angles

TRIGONOMETRY REVIEW. The Cosines and Sines of the Standard Angles TRIGONOMETRY REVIEW The Cosines and Sines of the Standad Angles P θ = ( cos θ, sin θ ) . ANGLES AND THEIR MEASURE In ode to define the tigonometic functions so that they can be used not only fo tiangula

More information

11.5 Graphs of Polar Equations

11.5 Graphs of Polar Equations 9 Applications of Tigonomet.5 Gaphs of Pola Equations In this section, we discuss how to gaph equations in pola coodinates on the ectangula coodinate plane. Since an given point in the plane has infinitel

More information

Core Maths C3. Revision Notes

Core Maths C3. Revision Notes Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...

More information

Review of Vectors. Appendix A A.1 DESCRIBING THE 3D WORLD: VECTORS. 3D Coordinates. Basic Properties of Vectors: Magnitude and Direction.

Review of Vectors. Appendix A A.1 DESCRIBING THE 3D WORLD: VECTORS. 3D Coordinates. Basic Properties of Vectors: Magnitude and Direction. Appendi A Review of Vectos This appendi is a summa of the mathematical aspects of vectos used in electicit and magnetism. Fo a moe detailed intoduction to vectos, see Chapte 1. A.1 DESCRIBING THE 3D WORLD:

More information

In the lecture on double integrals over non-rectangular domains we used to demonstrate the basic idea

In the lecture on double integrals over non-rectangular domains we used to demonstrate the basic idea Double Integals in Pola Coodinates In the lectue on double integals ove non-ectangula domains we used to demonstate the basic idea with gaphics and animations the following: Howeve this paticula example

More information

opp (the cotangent function) cot θ = adj opp Using this definition, the six trigonometric functions are well-defined for all angles

opp (the cotangent function) cot θ = adj opp Using this definition, the six trigonometric functions are well-defined for all angles Definition of Trigonometric Functions using Right Triangle: C hp A θ B Given an right triangle ABC, suppose angle θ is an angle inside ABC, label the leg osite θ the osite side, label the leg acent to

More information

Algebra and Trig. I. A point is a location or position that has no size or dimension.

Algebra and Trig. I. A point is a location or position that has no size or dimension. Algeba and Tig. I 4.1 Angles and Radian Measues A Point A A B Line AB AB A point is a location o position that has no size o dimension. A line extends indefinitely in both diections and contains an infinite

More information

Modern Linear Algebra

Modern Linear Algebra Hochschule fü Witschaft und Recht Belin Belin School of Economics and Law Wintesemeste 04/05 D. Hon Mathematics fo Business and Economics LV-N. 0069.0 Moden Linea Algeba (A Geometic Algeba cash couse,

More information

Mechanics 1: Work, Power and Kinetic Energy

Mechanics 1: Work, Power and Kinetic Energy Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).

More information

D.3. Angles and Degree Measure. Review of Trigonometric Functions

D.3. Angles and Degree Measure. Review of Trigonometric Functions APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric

More information

Trigonometric Identities & Formulas Tutorial Services Mission del Paso Campus

Trigonometric Identities & Formulas Tutorial Services Mission del Paso Campus Tigonometic Identities & Fomulas Tutoial Sevices Mission del Paso Campus Recipocal Identities csc csc Ratio o Quotient Identities cos cot cos cos sec sec cos = cos cos = cot cot cot Pthagoean Identities

More information

In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.

In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. Radians At school we usually lean to measue an angle in degees. Howeve, thee ae othe ways of measuing an angle. One that we ae going to have a look at hee is measuing angles in units called adians. In

More information

Write and Graph Equations of Circles

Write and Graph Equations of Circles 0.7 Wite and Gaph Equations of icles Befoe You wote equations of lines in the coodinate plane. Now You will wite equations of cicles in the coodinate plane. Wh? So ou can detemine zones of a commute sstem,

More information

Transformations in Homogeneous Coordinates

Transformations in Homogeneous Coordinates Tansfomations in Homogeneous Coodinates (Com S 4/ Notes) Yan-Bin Jia Aug, 6 Homogeneous Tansfomations A pojective tansfomation of the pojective plane is a mapping L : P P defined as u a b c u au + bv +

More information

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

More information

Mechanics 1: Motion in a Central Force Field

Mechanics 1: Motion in a Central Force Field Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.

More information

Chapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6

Chapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6 Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe

More information

Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!

Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360! 1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the

More information

Chapter 3: Vectors and Coordinate Systems

Chapter 3: Vectors and Coordinate Systems Coodinate Systems Chapte 3: Vectos and Coodinate Systems Used to descibe the position of a point in space Coodinate system consists of a fied efeence point called the oigin specific aes with scales and

More information

Unit Vectors. the unit vector rˆ. Thus, in the case at hand, 5.00 rˆ, means 5.00 m/s at 36.0.

Unit Vectors. the unit vector rˆ. Thus, in the case at hand, 5.00 rˆ, means 5.00 m/s at 36.0. Unit Vectos What is pobabl the most common mistake involving unit vectos is simpl leaving thei hats off. While leaving the hat off a unit vecto is a nast communication eo in its own ight, it also leads

More information

Pre-Calculus II. where 1 is the radius of the circle and t is the radian measure of the central angle.

Pre-Calculus II. where 1 is the radius of the circle and t is the radian measure of the central angle. Pre-Calculus II 4.2 Trigonometric Functions: The Unit Circle The unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this unit circle

More information

Section 9.4 Trigonometric Functions of any Angle

Section 9.4 Trigonometric Functions of any Angle Section 9. Trigonometric Functions of any Angle So far we have only really looked at trigonometric functions of acute (less than 90º) angles. We would like to be able to find the trigonometric functions

More information

Functions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem

Functions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem Intoduction One Function of Random Vaiables Functions of a Random Vaiable: Density Math 45 Into to Pobability Lectue 30 Let gx) = y be a one-to-one function whose deiatie is nonzeo on some egion A of the

More information

STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION

STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts

More information

The Binomial Distribution

The Binomial Distribution The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between

More information

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses, 3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

More information

Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapter 3 Savings, Present Value and Ricardian Equivalence Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

More information

Core Maths C2. Revision Notes

Core Maths C2. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

More information

Solutions to Homework Set #5 Phys2414 Fall 2005

Solutions to Homework Set #5 Phys2414 Fall 2005 Solution Set #5 1 Solutions to Homewok Set #5 Phys414 Fall 005 Note: The numbes in the boxes coespond to those that ae geneated by WebAssign. The numbes on you individual assignment will vay. Any calculated

More information

Samples of conceptual and analytical/numerical questions from chap 21, C&J, 7E

Samples of conceptual and analytical/numerical questions from chap 21, C&J, 7E CHAPTER 1 Magnetism CONCEPTUAL QUESTIONS Cutnell & Johnson 7E 3. ssm A chaged paticle, passing though a cetain egion of space, has a velocity whose magnitude and diection emain constant, (a) If it is known

More information

Week 3-4: Permutations and Combinations

Week 3-4: Permutations and Combinations Week 3-4: Pemutations and Combinations Febuay 24, 2016 1 Two Counting Pinciples Addition Pinciple Let S 1, S 2,, S m be disjoint subsets of a finite set S If S S 1 S 2 S m, then S S 1 + S 2 + + S m Multiplication

More information

Summary: Vectors. This theorem is used to find any points (or position vectors) on a given line (direction vector). Two ways RT can be applied:

Summary: Vectors. This theorem is used to find any points (or position vectors) on a given line (direction vector). Two ways RT can be applied: Summ: Vectos ) Rtio Theoem (RT) This theoem is used to find n points (o position vectos) on given line (diection vecto). Two ws RT cn e pplied: Cse : If the point lies BETWEEN two known position vectos

More information

Math Placement Test Practice Problems

Math Placement Test Practice Problems Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211

More information

92.131 Calculus 1 Optimization Problems

92.131 Calculus 1 Optimization Problems 9 Calculus Optimization Poblems ) A Noman window has the outline of a semicicle on top of a ectangle as shown in the figue Suppose thee is 8 + π feet of wood tim available fo all 4 sides of the ectangle

More information

3. EVALUATION OF TRIGONOMETRIC FUNCTIONS

3. EVALUATION OF TRIGONOMETRIC FUNCTIONS . EVALUATIN F TIGNMETIC FUNCTINS In this section, we obtain values of the trigonometric functions for quadrantal angles, we introduce the idea of reference angles, and we discuss the use of a calculator

More information

Semipartial (Part) and Partial Correlation

Semipartial (Part) and Partial Correlation Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated

More information

Graphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities.

Graphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities. Gaphs of Equations CHAT Pe-Calculus A coodinate sstem is a wa to gaphicall show the elationship between quantities. Definition: A solution of an equation in two vaiables and is an odeed pai (a, b) such

More information

Review Module: Dot Product

Review Module: Dot Product MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics 801 Fall 2009 Review Module: Dot Poduct We shall intoduce a vecto opeation, called the dot poduct o scala poduct that takes any two vectos and

More information

Th Po er of th Cir l. Lesson3. Unit UNIT 6 GEOMETRIC FORM AND ITS FUNCTION

Th Po er of th Cir l. Lesson3. Unit UNIT 6 GEOMETRIC FORM AND ITS FUNCTION Lesson3 Th Po e of th Ci l Quadilateals and tiangles ae used to make eveyday things wok. Right tiangles ae the basis fo tigonometic atios elating angle measues to atios of lengths of sides. Anothe family

More information

New proofs for the perimeter and area of a circle

New proofs for the perimeter and area of a circle New poofs fo the peimete and aea of a cicle K. Raghul Kuma Reseach Schola, Depatment of Physics, Nallamuthu Gounde Mahalingam College, Pollachi, Tamil Nadu 64001, India 1 aghul_physics@yahoo.com aghulkumak5@gmail.com

More information

Right Triangle Trigonometry

Right Triangle Trigonometry Right Triangle Trigonometry MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: evaluate trigonometric functions of acute angles, use

More information

K.S.E.E.B., Malleshwaram, Bangalore SSLC Mathematics-Model Question Paper-1 (2015) Regular Private Candidates (New Syllabus)

K.S.E.E.B., Malleshwaram, Bangalore SSLC Mathematics-Model Question Paper-1 (2015) Regular Private Candidates (New Syllabus) K.S.E.E.B., Malleshwaam, Bangaloe SSLC Mathematics-Model Question Pape-1 (015) Regula Pivate Candidates (New Syllabus) Max Maks: 100 No. of Questions: 50 Time: 3 Hous Code No. : Fou altenatives ae given

More information

Section 5-9 Inverse Trigonometric Functions

Section 5-9 Inverse Trigonometric Functions 46 5 TRIGONOMETRIC FUNCTIONS Section 5-9 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions

More information

Trigonometric Functions: The Unit Circle

Trigonometric Functions: The Unit Circle Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry

More information

3 Unit Circle Trigonometry

3 Unit Circle Trigonometry 0606_CH0_-78.QXP //0 :6 AM Page Unit Circle Trigonometr In This Chapter. The Circular Functions. Graphs of Sine and Cosine Functions. Graphs of Other Trigonometric Functions. Special Identities.5 Inverse

More information

Review of Coordinate Systems

Review of Coordinate Systems Review o Coodinate Sstems good undestanding o coodinate sstems can be ve helpul in solving poblems elated to Mawell s Equations. The thee most common coodinate sstems ae ectangula (,, ), clindical (,,

More information

CHAPTER 10 Aggregate Demand I

CHAPTER 10 Aggregate Demand I CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income

More information

Continuous Compounding and Annualization

Continuous Compounding and Annualization Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem

More information

5.2. Trigonometric Functions Of Real Numbers. Copyright Cengage Learning. All rights reserved.

5.2. Trigonometric Functions Of Real Numbers. Copyright Cengage Learning. All rights reserved. 5.2 Trigonometric Functions Of Real Numbers Copyright Cengage Learning. All rights reserved. Objectives The Trigonometric Functions Values of the Trigonometric Functions Fundamental Identities 2 Trigonometric

More information

10.5 Graphs of the Trigonometric Functions

10.5 Graphs of the Trigonometric Functions 790 Foundations of Trigonometr 0.5 Graphs of the Trigonometric Functions In this section, we return to our discussion of the circular trigonometric functions as functions of real numbers and pick up where

More information

We d like to explore the question of inverses of the sine, tangent, and secant functions. We ll start with f ( x)

We d like to explore the question of inverses of the sine, tangent, and secant functions. We ll start with f ( x) Inverse Trigonometric Functions: We d like to eplore the question of inverses of the sine, tangent, and secant functions. We ll start with f ( ) sin. Recall the graph: MATH 30 Lecture 0 of 0 Well, we can

More information

θ = 45 θ = 135 θ = 225 θ = 675 θ = 45 θ = 135 θ = 225 θ = 675 Trigonometry (A): Trigonometry Ratios You will learn:

θ = 45 θ = 135 θ = 225 θ = 675 θ = 45 θ = 135 θ = 225 θ = 675 Trigonometry (A): Trigonometry Ratios You will learn: Trigonometr (A): Trigonometr Ratios You will learn: () Concept of Basic Angles () how to form simple trigonometr ratios in all 4 quadrants () how to find the eact values of trigonometr ratios for special

More information

Trigonometric Functions

Trigonometric Functions LIALMC5_1768.QXP /6/4 1:4 AM Page 47 5 Trigonometric Functions Highwa transportation is critical to the econom of the United States. In 197 there were 115 billion miles traveled, and b the ear this increased

More information

PY1052 Problem Set 8 Autumn 2004 Solutions

PY1052 Problem Set 8 Autumn 2004 Solutions PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what

More information

Power and Sample Size Calculations for the 2-Sample Z-Statistic

Power and Sample Size Calculations for the 2-Sample Z-Statistic Powe and Sample Size Calculations fo the -Sample Z-Statistic James H. Steige ovembe 4, 004 Topics fo this Module. Reviewing Results fo the -Sample Z (a) Powe and Sample Size in Tems of a oncentality Paamete.

More information

Review of Essential Skills and Knowledge

Review of Essential Skills and Knowledge Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope

More information

est using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.

est using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years. 9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,

More information

1.4 Phase Line and Bifurcation Diag

1.4 Phase Line and Bifurcation Diag Dynamical Systems: Pat 2 2 Bifucation Theoy In pactical applications that involve diffeential equations it vey often happens that the diffeential equation contains paametes and the value of these paametes

More information

Radian Measure and Dynamic Trigonometry

Radian Measure and Dynamic Trigonometry cob980_ch0_089-.qd 0//09 7:0 Page 89 Debd MHDQ-New:MHDQ:MHDQ-.: CHAPTER CONNECTIONS Radian Measue and Dnamic Tigonomet CHAPTER OUTLINE. Angle Measue in Radians 90. Ac Length, Velocit, and the Aea of a

More information

4.6 GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS. Copyright Cengage Learning. All rights reserved.

4.6 GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS. Copyright Cengage Learning. All rights reserved. 4.6 GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch the graphs of tangent functions. Sketch the graphs of cotangent functions. Sketch

More information

In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.

In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. Radians mc-ty-adians-2009-1 Atschoolweusuallyleantomeasueanangleindegees. Howeve,theeaeothewaysof measuinganangle. Onethatweaegoingtohavealookatheeismeasuinganglesinunits called adians. In many scientific

More information

On Correlation Coefficient. The correlation coefficient indicates the degree of linear dependence of two random variables.

On Correlation Coefficient. The correlation coefficient indicates the degree of linear dependence of two random variables. C.Candan EE3/53-METU On Coelation Coefficient The coelation coefficient indicates the degee of linea dependence of two andom vaiables. It is defined as ( )( )} σ σ Popeties: 1. 1. (See appendi fo the poof

More information

Angles and Their Measure

Angles and Their Measure Trigonometry Lecture Notes Section 5.1 Angles and Their Measure Definitions: A Ray is part of a line that has only one end point and extends forever in the opposite direction. An Angle is formed by two

More information

y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx

y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.

More information

11 Trigonometric Functions of Acute Angles

11 Trigonometric Functions of Acute Angles Arkansas Tech University MATH 10: Trigonometry Dr. Marcel B. Finan 11 Trigonometric Functions of Acute Angles In this section you will learn (1) how to find the trigonometric functions using right triangles,

More information

Physics 505 Homework No. 5 Solutions S5-1. 1. Angular momentum uncertainty relations. A system is in the lm eigenstate of L 2, L z.

Physics 505 Homework No. 5 Solutions S5-1. 1. Angular momentum uncertainty relations. A system is in the lm eigenstate of L 2, L z. Physics 55 Homewok No. 5 s S5-. Angula momentum uncetainty elations. A system is in the lm eigenstate of L 2, L z. a Show that the expectation values of L ± = L x ± il y, L x, and L y all vanish. ψ lm

More information

y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx y 1 u 2 du u 1 3u 3 C

y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx y 1 u 2 du u 1 3u 3 C Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.

More information

NURBS Drawing Week 5, Lecture 10

NURBS Drawing Week 5, Lecture 10 CS 43/585 Compute Gaphics I NURBS Dawing Week 5, Lectue 1 David Been, William Regli and Maim Pesakhov Geometic and Intelligent Computing Laboato Depatment of Compute Science Deel Univesit http://gicl.cs.deel.edu

More information

Hour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and

Hour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and Hou Exam No. Please attempt all of the following poblems befoe the due date. All poblems count the same even though some ae moe complex than othes. Assume that c units ae used thoughout. Poblem A photon

More information

Moment and couple. In 3-D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r

Moment and couple. In 3-D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r Moment and couple In 3-D, because the detemination of the distance can be tedious, a vecto appoach becomes advantageous. o k j i M k j i M o ) ( ) ( ) ( + + M o M + + + + M M + O A Moment about an abita

More information

4.3 & 4.8 Right Triangle Trigonometry. Anatomy of Right Triangles

4.3 & 4.8 Right Triangle Trigonometry. Anatomy of Right Triangles 4.3 & 4.8 Right Triangle Trigonometry Anatomy of Right Triangles The right triangle shown at the right uses lower case a, b and c for its sides with c being the hypotenuse. The sides a and b are referred

More information

Notes on Electric Fields of Continuous Charge Distributions

Notes on Electric Fields of Continuous Charge Distributions Notes on Electic Fields of Continuous Chage Distibutions Fo discete point-like electic chages, the net electic field is a vecto sum of the fields due to individual chages. Fo a continuous chage distibution

More information

The Detection of Obstacles Using Features by the Horizon View Camera

The Detection of Obstacles Using Features by the Horizon View Camera The Detection of Obstacles Using Featues b the Hoizon View Camea Aami Iwata, Kunihito Kato, Kazuhiko Yamamoto Depatment of Infomation Science, Facult of Engineeing, Gifu Univesit aa@am.info.gifu-u.ac.jp

More information

2. SCALARS, VECTORS, TENSORS, AND DYADS

2. SCALARS, VECTORS, TENSORS, AND DYADS 2. SCALARS, VECTORS, TENSORS, AND DYADS This section is a eview of the popeties of scalas, vectos, and tensos. We also intoduce the concept of a dyad, which is useful in MHD. A scala is a quantity that

More information

Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus

Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:

More information

9. Mathematics Practice Paper for Class XII (CBSE) Available Online Tutoring for students of classes 4 to 12 in Physics, Chemistry, Mathematics

9. Mathematics Practice Paper for Class XII (CBSE) Available Online Tutoring for students of classes 4 to 12 in Physics, Chemistry, Mathematics Available Online Tutoing fo students of classes 4 to 1 in Physics, 9. Mathematics Class 1 Pactice Pape 1 3 1. Wite the pincipal value of cos.. Wite the ange of the pincipal banch of sec 1 defined on the

More information

On Some Functions Involving the lcm and gcd of Integer Tuples

On Some Functions Involving the lcm and gcd of Integer Tuples SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. MATH. INFORM. AND MECH. vol. 6, 2 (2014), 91-100. On Some Functions Involving the lcm and gcd of Intege Tuples O. Bagdasa Abstact:

More information

2.1 Three Dimensional Curves and Surfaces

2.1 Three Dimensional Curves and Surfaces . Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two- or three-dimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The

More information

CLOSE RANGE PHOTOGRAMMETRY WITH CCD CAMERAS AND MATCHING METHODS - APPLIED TO THE FRACTURE SURFACE OF AN IRON BOLT

CLOSE RANGE PHOTOGRAMMETRY WITH CCD CAMERAS AND MATCHING METHODS - APPLIED TO THE FRACTURE SURFACE OF AN IRON BOLT CLOSE RANGE PHOTOGRAMMETR WITH CCD CAMERAS AND MATCHING METHODS - APPLIED TO THE FRACTURE SURFACE OF AN IRON BOLT Tim Suthau, John Moé, Albet Wieemann an Jens Fanzen Technical Univesit of Belin, Depatment

More information

1. The Six Trigonometric Functions 1.1 Angles, Degrees, and Special Triangles 1.2 The Rectangular Coordinate System 1.3 Definition I: Trigonometric

1. The Six Trigonometric Functions 1.1 Angles, Degrees, and Special Triangles 1.2 The Rectangular Coordinate System 1.3 Definition I: Trigonometric 1. The Si Trigonometric Functions 1.1 Angles, Degrees, and Special Triangles 1. The Rectangular Coordinate Sstem 1.3 Definition I: Trigonometric Functions 1.4 Introduction to Identities 1.5 More on Identities

More information

Model Question Paper Mathematics Class XII

Model Question Paper Mathematics Class XII Model Question Pape Mathematics Class XII Time Allowed : 3 hous Maks: 100 Ma: Geneal Instuctions (i) The question pape consists of thee pats A, B and C. Each question of each pat is compulsoy. (ii) Pat

More information

Lesson Plan. Students will be able to define sine and cosine functions based on a right triangle

Lesson Plan. Students will be able to define sine and cosine functions based on a right triangle Lesson Plan Header: Name: Unit Title: Right Triangle Trig without the Unit Circle (Unit in 007860867) Lesson title: Solving Right Triangles Date: Duration of Lesson: 90 min. Day Number: Grade Level: 11th/1th

More information

CLASS XI CHAPTER 3. Theorem 1 (sine formula) In any triangle, sides are proportional to the sines of the opposite angles. That is, in a triangle ABC

CLASS XI CHAPTER 3. Theorem 1 (sine formula) In any triangle, sides are proportional to the sines of the opposite angles. That is, in a triangle ABC CLASS XI Anneue I CHAPTER.6. Poofs and Simple Applications of sine and cosine fomulae Let ABC be a tiangle. By angle A we mean te angle between te sides AB and AC wic lies between 0 and 80. Te angles B

More information

Right Triangle Trigonometry

Right Triangle Trigonometry Section 6.4 OBJECTIVE : Right Triangle Trigonometry Understanding the Right Triangle Definitions of the Trigonometric Functions otenuse osite side otenuse acent side acent side osite side We will be concerned

More information

Trigonometry LESSON TWO - The Unit Circle Lesson Notes

Trigonometry LESSON TWO - The Unit Circle Lesson Notes (cosθ, sinθ) Trigonometry Example 1 Introduction to Circle Equations. a) A circle centered at the origin can be represented by the relation x 2 + y 2 = r 2, where r is the radius of the circle. Draw each

More information

Chapter 5: Trigonometric Functions of Real Numbers

Chapter 5: Trigonometric Functions of Real Numbers Chapter 5: Trigonometric Functions of Real Numbers 5.1 The Unit Circle The unit circle is the circle of radius 1 centered at the origin. Its equation is x + y = 1 Example: The point P (x, 1 ) is on the

More information