4.1  Trigonometric Functions of Acute Angles


 Jared Potter
 1 years ago
 Views:
Transcription
1 4.1  Tigonometic Functions of cute ngles a is a halfline 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 a as the initial a and the othe a as the teminal a, the measue of the angle is the amount of otation needed to make the initial a coincide with the teminal a. common unit of the measue of an angle is a degee. We define one degee (1 ) to be 1/60 of a complete counteclockwise otation. Thee ae 60 in a complete counteclockwise otation. It is common to use the Geek lette (theta) to epesent an angle measue. n acute angle measues between 0 and 90, and an obtuse angle measues between 90 and180. ight tiangle is a tiangle that contains a90 angle. pg. 179 Definition of the Tigonometic Functions If θ (theta) is an acute angle in a ight tiangle, then Name of Function bbeviation Ratio sine of angle θ sin θ opposite hpotenuse cosecant of angle θ csc θ hpotenuse opposite cosine of angle θ secant of angle θ tangent of angle θ cotangent of angle θ cos θ sec θ tan θ cot θ adjacent hpotenuse hpotenuse adjacent opposite adjacent adjacent opposite Pogess Check Find the values of the emaining tigonometic functions of the acute angle θ if cos. 5 Pogess Check In tiangle BC with C 90, if 8 sin and a 40 find c. 17 1
2 pg. 00 Eact Tigonometic Values fo 0, 45, and 60 θ Sin θ Csc θ Cos θ Sec θ Tan θ Cot θ tiangle tiangle Pogess Check 4 Find the eact values of sin 60, cot 0, csc0, and cos 45 pg. 01 Cofunction Popeties Fo an acute angle θ, sin(90 θ) = cos θ tan(90 θ) = cot θ sec(90 θ) = csc θ cos(90 θ) = sin θ cot(90 θ) = tan θ csc(90 θ) = sec θ Pogess Check 5 Epess cos4 as a function of the acute angle complimenta to 4. Pogess Check 6 Evaluate each epession b calculato to fou decimal places. a. tan 6 b. sec16.7 c. sin81 50 Pogess Check 8 Use a calculato to appoimate the acute angle that satisfies the given equation. Wite solutions to the neaest tenth of a degee and to the neaest 10 minutes. a. cos 0.78 b. csc 1.448
3 4.  Right Tiangle pplications To solve ight tiangle means to find the measues of the two acute angles and the length of the thee sides of the tiangle. To accomplish this, at least two of these five values must be known, and one o moe must be a side length. How to Round when Solving a Right Tiangle ccuac of Sides Two significant digits Thee significant digits Fou significant digits ccuac of ngles Neaest degee Neaest 10 minutes o tenth of a degee Neaest minute o hundedth of a degee Diagam of an BC Right Tiangle Solving a Right Tiangle: ngleside Case Pogess Check Solve the ight tiangle BC in which 9 40' and c 7.5 ft.
4 Solving a Right Tiangle: Two Sides Case Pogess Check Solve the ight tiangle BC in which b 1.0 ft. and c 19.0 ft. Pogess Check 4 ladde leans against the side of a building and makes an angle of 7.0 with the gound. If the ladde is 5.0 ft. long, then find the height the ladde eaches on the building. Pogess Check 5 suveo stands on a cliff 175 ft. above a ive. If the angle of depession to the ive s edge on the opposite bank is8.4, how wide is the ive at this point? 4
5 4.  Tigonometic Functions of Geneal ngles pg. 14 Definition of the Tigonometic Functions If θ (theta) is an angle in standad position, and if (, ) is an point on the teminal a of θ [ecept (0,0)], then Name of Function bbeviation Ratio sine of angle θ sin θ cosecant of angle θ csc θ ( 0) cosine of angle θ cos θ secant of angle θ sec θ ( 0) tangent of angle θ tan θ cotangent of angle θ cot θ ( 0) Pogess Check 1 Find the values of the si tigonometic functions of angle if, 1 is a point on the teminal a of. 5
6 Pogess Check Find the values of the emaining tigonometic functions if sin and tan 0. 4 Quadtantal ngles ae angles that have a teminal a that coincides with one of the aes. n quadantal angle can be epessed as the poduct of 90 o and some intege. In geneal, if two angles have the same teminal a, the ae called coteminal, and the tigonometic functions of coteminal angles ae equal. pg. 17 θ Sin θ Csc θ Cos θ Sec θ Tan θ Cot θ 0 0 Undefined Undefined Undefined Undefined Undefined Undefined Undefined Undefined 0 Refeence angles ae useful in evaluating tigonometic functions fo angles that ae not quadantal angles. The efeence angle fo an angle θ is defined to be the positive acute angle fomed b the teminal a of θ and the hoizontal ais. To Evaluate Tigonometic Functions Fo nonquadantal angles: 1. Find the efeence angle fo the given angle.. Find the tigonometic value of the efeence angle using the appopiate function. If the efeence angle is 0 o, 45 o, o 60 o, the eact answe is pefeable.. Detemine the coect sign accoding to the teminal a of the angle. 6
7 Pogess Check Find the si tigonometic functions of 180, using the definition of the tigonometic functions. Pogess Check 4 Find the eact value of each epession. a. cos70 b. cot 90 Pogess Check 5 Find the eact value of sin00. Pogess Check 6 Find the eact value of sin15. 7
8 4.4  Intoduction to Tigonometic Equations pg. 5 Quadant Solution 1 efeence angle 180 o efeence angle 180 o + efeence angle 4 60 o efeence angle Pogess Check Find the eact values of 0 60 tue statement. fo which the equation cos 1 0 is a 8
9 Pogess Check To the neaest 10 minutes, appoimate the values of 0 60 equation sin is a tue statement. fo which the Pogess Check 4 Solve tan 5 fo Round off answes to the neaest tenth of a degee. Pogess Check 5 ppoimate all the solutions to sec 7 1, to the neaest 10 minutes. 9
10 Chapte 4 Oveall Review atios sin cos tan O H H O csc = sec cot H O H O a b c USE WHEN: 1) Given diagam of a ight tiangle with side lengths find ) Given one tig atio find emaining tig atios *If not pefect squae oot leave H O sides find c B SOLVING RIGHT TRINGLE BC : 1) Given one angle and one side find one angle and two a ) Given two sides find one side and two angles ) Given one tig atio and one side of simila tiangle anothe side of the simila tiangle *If sides ae given with significant digits, angles must be to the neaest degee. C *If sides ae given with significant digits, b angles must be to the neaest tenth of a degee o neaest ten minutes 10
11 sin cos tan csc = sec cot o USE WHEN: 1) Given a point on the teminal a ) Given an equation and quadant ) Given one tig atio and quadant NOTE: and ma be negative depending what quadant the teminal a is in, but is neve negative. *If not pefect squae oot leave S T C b Finding the values of 0 60g fo which a tigonometic equation is a tue statement: 1) Solve fo the tigonometic function (if necessa). ) Detemine which two quadants contain the teminal a of. ) Detemine the efeence angle: * Use tig invese buttons on calculato to find the efeence angle but don t use negative sign if tig atio has one. 4) Detemine the two appopiate values of b dawing o use the following, Q1: Ref Q : 180 Ref Q: 180 Ref Q : 60 Ref 4 Chapte 9 Law of Cosines a b c bc Cos Law of Sines sin sin B sinc a b c 11
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 informationChapter 5.3: Circular Trigonometric Functions
Chapte 5.3: Cicula Tigonometic Functions A efeence tiangle is fomed b dopping a pependicula (altitude) fom the teminal a of a standad position angle to the ais, that is, again, the ais. The efeence angle
More informationUNIT 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 informationTrigonometry 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 informationAngles in Standard Positions Lesson Plan
Angles in Standad Positions Lesson Plan B: Douglas A. Rub Date: 10/10/00 Class: PeCalculus II Gades: 11/1 INSTRUCTIONAL OBJECTIVES: At the end of this lesson, the student will be able to: 1. Coectl identif
More informationTrigonometric Functions of Any Angle. cos. sin r. cot, r csc, y. y 0
0_00.qd 1 1/7/05 Chapte. 11:05 AM Page 1 Tigonomet Tigonometic Functions of An Angle What ou should lean Evaluate tigonometic functions of an angle. Use efeence angles to evaluate tigonometic functions.
More informationSection 5.2: Trigonometric Functions of Angles
Section 5.: Tigonometic Functions of Angles Objectives: Upon completion of this lesson, ou will be able to: find the values of the si tigonometic functions fo an angle with given conditions given an angle
More information2.2. 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 informationReviewing Trigonometry
Reviewing Tigonomet Intoduction and Definition Students often think the ae unpepaed fo Calculus if the ae not full comfotable with tigonomet. Howeve, tigonomet is not in an wa the mateial that pecedes
More information2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES
. 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
More informationBasic Trigonometry ( )
PeCalculus Assignment Sheet Unit : Basic Tigonomety Septembe th Octobe 11 th, 01 Date Topic Assignment Tuesday (.1) Angles in Coodinate Plane Degees p.1 #1 all / Notes pages 1 and Wednesday Pape Plate
More informationUnit Circle Lesson Plan
Unit Cicle Lesson Plan B: Douglas A. Rub Date: 0/0/2002 Class: PeCalculus Gades: /2 NSTRUCTONAL OBJECTVES: At the end of this lesson, the student will be able to:. Given a eal numbe that is an integal
More informationSection 5.2: Trigonometric Functions of Angles
Section 5.: Tigonometic Functions of Angles Objectives Upon completion of this lesson, ou will be able to: Find the values of the si tigonometic functions fo an angle θ with given conditions. Given an
More informationAlgebra III: Hutschenreuter Chapter 7 Trigonometric Functions. Section 7.1: Measurement of Angles
Algeba III: Hutscheneute Chapte 7 Tigonometic Functions Essential Question: Section 7.1: Measuement of Angles Name Date Tigonomety Comes fom two Geek wods TRIGONON and METRON meaning tiangle measuement
More informationOriginally 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 informationAlgebra 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 informationCHAT PreCalculus Section 10.7. Polar Coordinates
CHAT PeCalculus 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 information4.3 Right Triangle Trigonometry
. Right Triangle Trigonometr Section. Notes Page This is a ver important section since we are giving definitions for the si trigonometric functions ou be using throughout the rest of this course and beond.
More informationTrigonometric 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 informationName Period 11/14 11/18 GL
Name Peiod 11/14 11/18 GL UNIT 8  RADICALS AND PYTHAGOREAN THEOREM I can define, identify and illustate the following tems leg of a ight tiangle adical squae oot Radicand hypotenuse Pythagoean theoem
More informationTransition to College Math
Transition to College Math Date: Unit 3: Trigonometr Lesson 2: Angles of Rotation Name Period Essential Question: What is the reference angle for an angle of 15? Standard: FTF.2 Learning Target: Eplain
More informationVECTOR MECHANICS FOR ENGINEERS: Statics of Particles. J. Walt Oler The McGrawHill Companies, Inc. All rights reserved.
VECTOR MECHANICS FOR ENGINEERS: STATICS Statics of Paticles J. Walt Ole Teas Tech Univesit 2007 The McGawHill Companies, Inc. All ights eseved. Vecto Mechanics fo Enginees: Statics Contents Intoduction
More informationSkills 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 informationopp (the cotangent function) cot θ = adj opp Using this definition, the six trigonometric functions are welldefined 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 informationTRIGONOMETRY 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 informationSection 53 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 informationMaking Pi. Then we automatically get the formula C = 2πr, which enables us to evaluate C whenever we know the value of r.
Maths 1 Extension Notes #3.b Not Examinable Making Pi 1 Intoduction 1.1 Definition of Pi Conside any cicle with adius and cicumfeence C. In the following sections, we show that the atio C 2 is just a numbe
More informationChapter 4. Trigonometric Functions. Selected Applications
Chapte Tigonometic Functions. Radian and Degee Measue. Tigonometic Functions: The Unit Cicle. Right Tiangle Tigonomet. Tigonometic Functions of An Angle. Gaphs of Sine and Cosine Functions. Gaphs of Othe
More information8.4 Torque. Torque. Rotational Dynamics. ProblemSolving
8.4 oque oque otational Dynamics PoblemSolving We began this couse with chaptes on kinematics, the desciption of motion without asking about its causes. We then found that foces cause motion, and used
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
MAC Module Test Name MULTIPLE CHOICE. Choose the one altenative that best completes the statement o answes the question. Convet to ectangula coodinates. ), π (, 0) (0, ) C) (, 0) (0, ) Answe: D Objective:
More informationCopyright Cengage Learning. All rights reserved.
6 Trigonometry Copyright Cengage Learning. All rights reserved. 6.2 Right Triangle Trigonometry Copyright Cengage Learning. All rights reserved. Objectives Evaluate trigonometric functions of acute angles
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D. Review of Trigonometric Functions D7 APPENDIX D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving
More informationD.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 informationTANGENTS IN POLAR COORDINATES
TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS. Polacoodinate equations fo lines A pola coodinate system in the plane is detemined by a Pole P and a halfline called the pola
More informationSECTION TWO: TRIGONOMETRIC FUNCTIONS AND THEIR TABLES: Unit Circle: x 2 + y 2 = 1. P(x,y) 2 2
SECTION TWO: TRIGONOMETRIC FUNCTIONS ND THEIR TLES:. TRIGONOMETRIC FUNCTIONS ND THEIR TLES: Let P(,) be an point on the unit circle: Unit Circle: + = P(,)    If P is the terminal point that corresponds
More informationChapter 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 informationScale Drawings 1. Measure each side and angle, and sketch the polygon using the scale 1 cm represents 2.5 cm.
Chapte 6 Peequisite Skills BLM 61.. Scale Dawings 1. Measue each side and angle, and sketch the polygon using the scale 1 cm epesents 2.5 cm. 6. Use the cosine law to find the length of side s. 7. Use
More informationK.S.E.E.B., Malleshwaram, Bangalore SSLC MathematicsModel Question Paper1 (2015) Regular Private Candidates (New Syllabus)
K.S.E.E.B., Malleshwaam, Bangaloe SSLC MathematicsModel Question Pape1 (015) Regula Pivate Candidates (New Syllabus) Max Maks: 100 No. of Questions: 50 Time: 3 Hous Code No. : Fou altenatives ae given
More informationMath 144 Activity #3 Coterminal Angles and Reference Angles
Math 144 Activity #3 Coterminal Angles and Reference Angles For this activity we will be referring to the unit circle. Using the unit circle below, explain how you can find the sine of any given angle,
More information5.2. 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 informationTrigonometry Notes on Unit Circle Trigonometry.
Trigonometr Notes on Unit Circle Trigonometr. Epanding the Definition of the Trigonometric Functions: As time went b and the concept of angles epanded beond acute and obtuse angles, it was discovered that
More informationAngles and Degree Measure. Figure D.25 Figure D.26
APPENDIX D. Review of Trigonometric Functions D7 APPE N DIX D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions
More informationUnit 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 informationWarm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2
Graphs of of Other Trigonometric 142 Warm Up Lesson Presentation Lesson Quiz 2 Warm Up If sin A =, evaluate. 1. cos A 2. tan A 3. cot A 4. sec A 5. csc A Objective Recognize and graph trigonometric functions.
More informationRight 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 informationIn 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 informationTrigonometric (Polar) Form In the figure the complex number x + yi corresponds to a vector OP with direction angle θ and magnitude r.
1 of 5 8/6/2004 8.5 TRIGONOMETRIC (POLAR FORM OF 8.5 TRIGONOMETRIC (POLAR FORM OF COMPLEX NUMBERS; PRODUCTS AND QUOTIENTS The Complex Plane and Vecto Repesentation Tigonometic (Pola Fom Poducts of Complex
More information2. Trigonometric Functions
. Tigonometic Functions. Radians Definition A adian is a measue of angle size. It is defined by the diagam at the ight, that is, adian is the angle which subtends an ac of length in a cicle of adius. Example.How
More information4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first nonzero 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 informationSection 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 informationFigure 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 information4.1. Radian and Degree Measure. Angle. What you should learn. Why you should learn it
_.qd 8 /7/5 Chapte. : AM Tigonomet Radian and Degee Measue What ou should lean Page 8 Descibe angles. Use adian measue. Use degee measue. Use angles to model and solve eallife poblems. Wh ou should lean
More informationLINES AND TANGENTS IN POLAR COORDINATES
LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Polacoodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and
More information11 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 informationTrigonometric Functions: Unit Circle Approach. Copyright Cengage Learning. All rights reserved.
Trigonometric Functions: Unit Circle Approach Copyright Cengage Learning. All rights reserved. 5.4 More Trigonometric Graphs Copyright Cengage Learning. All rights reserved. Objectives Graphs of Tangent,
More informationRIGHT TRIANGLE TRIGONOMETRY
RIGHT TRIANGLE TRIGONOMETRY The word Trigonometry can be broken into the parts Tri, gon, and metry, which means Three angle measurement, or equivalently Triangle measurement. Throughout this unit, we will
More informationCoordinate 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 informationTrigonometry (Chapters 4 5) Sample Test #1 First, a couple of things to help out:
First, a couple of things to help out: Page 1 of 24 Use periodic properties of the trigonometric functions to find the exact value of the expression. 1. cos 2. sin cos sin 2cos 4sin 3. cot cot 2 cot Sin
More informationThe Tangent Function
Math 36 "Fall 08" 4.3 "Graphs of Tangent, Cotangent, Secant, and Cosecant Functions" Skills Objectives: * Graph basic tangent, cotangent, secant, and cosecant functions. * Determine the period of tangent,
More informationReview 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 informationCopyright Cengage Learning. All rights reserved.
4 Trigonometry Copyright Cengage Learning. All rights reserved. 4.2 Trigonometric Functions: The Unit Circle Copyright Cengage Learning. All rights reserved. Objectives Identify a unit circle and describe
More information9. 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 information1.6 Trigonometric Functions
Section.6 Tigonometic Functions 45.6 Tigonometic Functions What ou will lean about... Radian Measue Gahs of Tigonometic Functions Peiodicit Even and Odd Tigonometic Functions Tansfomations of Tigonometic
More informationTrigonometric Functions: Unit Circle Approach. Copyright Cengage Learning. All rights reserved.
Trigonometric Functions: Unit Circle Approach Copyright Cengage Learning. All rights reserved. 5.5 Inverse Trigonometric Functions and their Graphs Copyright Cengage Learning. All rights reserved. Objectives
More informationy = rsin! (opp) x = z cos! (adj) sin! = y z = The Other Trig Functions
MATH 7 Right Triangle Trig Dr. Neal, WKU Previously, we have seen the right triangle formulas x = r cos and y = rsin where the hypotenuse r comes from the radius of a circle, and x is adjacent to and y
More informationSection 7.2 Right Triangle Trigonometry
Section 7. Right Triangle Trigonometry These are the angles we will use most often in this class (as well as in your calculus classes in the future). You need to commit these angles to memory as soon as
More informationRight 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 informationTrigonometry Semester Exam Review
Blizzard Bag Complete each problem on notebook paper. This activity usually takes 3 days to complete in class. If you don t understand a problem, show that you made an attempt and star the problem so
More informationSHAPE, SPACE AND MEASURES
SHPE, SPCE ND MESURES Pupils should be taught to: Use coodinates in all fou quadants s outcomes, Yea 7 pupils should, fo example: Use, ead and wite, spelling coectly: ow, column, coodinates, oigin, xaxis,
More informationMA Lesson 19 Summer 2016 Angles and Trigonometric Functions
DEFINITIONS: An angle is defined as the set of points determined by two rays, or halflines, l 1 and l having the same end point O. An angle can also be considered as two finite line segments with a common
More informationAlg2  CH13 Practice Test
lg  H13 Practice Test Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Find the value of the sine, cosine, and tangent functions for θ where = 96, = 8,
More informationSection 7.2 Right Triangle Trigonometry
Section 7. Right Triangle Trigonometry These are the angles we will use most often in this class (as well as in your calculus classes in the future). You need to commit these angles to memory as soon as
More information4.3 & 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 information1 of 5 8/6/ TRIGONOMETRIC FUNCTIONS
1 of 5 8/6/2004 5.2 TRIGONOMETRIC FUNCTIONS 5.2 TRIGONOMETRIC FUNCTIONS Topics : Trigonometric functions Quadrantal angles Reciprocal Identities Signs and ranges of function values Pythagorean identities
More information4.6 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 informationTrigonometry Gateway Exam Questions
MATH 1 Trigonometry Fall 00 Circle one: 8:0 / 9: Ms Kracht Trigonometry Gateway Exam Questions Each version of the Gateway Exam will consist of 0 questions Many of the questions on each version of the
More informationTrigonometric Functions
Chapter 4 Trigonometric Functions Course Number Section 4.1 Radian and Degree Measure Objective: In this lesson ou learned how to describe an angle and to convert between degree and radian measure. Instructor
More informationPreCalculus II. where 1 is the radius of the circle and t is the radian measure of the central angle.
PreCalculus 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 informationVector 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 informationReview Topics Lawrence B. Rees You may make a single copy of this document for personal use without written permission.
Review Topics Lawence. Rees 2006. You ma make a single cop of this document fo pesonal use without witten pemission. R.1 Vectos I assume that ou have alead studied vectos in pevious phsics couses. If ou
More information10 Torque. Lab. What You Need To Know: Physics 211 Lab
b Lab 10 Toque What You Need To Know: F (a) F F Angula Systems Evey lab up to this point has dealt with objects moving in the linea system. In othe wods, objects moving in a staight line. Now we ae going
More informationDETAILED SOLUTIONS AND CONCEPTS  THE SIX TRIGONOMETRIC RATIOS
DETAILED SOLUTIONS AND CONCEPTS  THE SIX TRIGONOMETRIC RATIOS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE
More informationClosure: In closing, be sure students understand the questions they missed and review those concepts again using the unit circle.
Name: Mendi White Grade Level/Subject: Trigonometry Topic: Unit Circle, Trigonometric Functions Objectives (P.A.S.S.): Introduction: This game is great for reviewing the unit circle and the basic aspects
More informationSECTION 58 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions
8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions 9 duce a scatter plot in the viewing window. Choose 8 for the viewing window. (B) It appears that a sine curve of the form k
More informationFind the exact values of the indicated trigonometric functions. Write fractions in lowest terms. 1) 15
MATH 170 Exam 1 Review Problems Name Review your notes and homework problems in addition to this worksheet. The exam covers 6.16., 8.1, 8. Find the exact values of the indicated trigonometric functions.
More information13 Trigonometric Functions of Real Numbers
Arkansas Tech University MATH 0: Trigonometry Dr. Marcel B. Finan Trigonometric Functions of Real Numbers In this section, you will () study the trigonometric functions of real numbers, () their properties,
More information13.1Introduction to Trigonometry Algebra 2
13.1Introduction to Trigonometry Algebra 2 Goal 1: Find values of trigonometric functions for acute angles. Goal 2: Solve problems involving right triangles. B c a A θ b C Trigonometric functions are ratios
More informationNew 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 informationSec 2 Hon Notes RIGHT TRIANGLE Trigonometry 8.1. adjacent _ leg cos hypotenuse
Sec Hon Notes RIGHT TRIANGLE Trigonometry 8.1 Trig Ratios: sine, cosine, tangent, cosecant, secant, cotangent (theta): acute angle opposite _ leg sin hypotenuse adjacent _ leg cos hypotenuse opposite _
More informationExploration of the Trigonometric Identities using the Unit Circle by Christine Kasitz
Grade level: 912 Exploration of the Trigonometric by Christine Kasitz Activity overview Students will investigate the relationship of the trigonometric functions to similar triangles created using the
More informationSECTION 4.1. Basic Graphs. Copyright Cengage Learning. All rights reserved.
SECTION 4.1 Basic Graphs Copyright Cengage Learning. All rights reserved. Learning Objectives 1 2 3 4 Sketch the graph of a basic trigonometric function. Analyze the graph of a trigonometric function.
More informationAlgebra 2B: Trigonometry Unit Test practice Part 1 Without Calculator, Part 2 with Calculator
Name: Class: Date: Algebra 2B: Trigonometry Unit Test practice Part 1 Without Calculator, Part 2 with Calculator Short Answer 1. Determine whether the function shown below is or is not periodic. If it
More information9.2 Angles of Rotation
9.2 Angles of Rotation Term Initial side Terminal side Positive s Negative s Coterminal s Angle of rotation Standard position Definition Where the rotation starts The ra on the positive  ais Also referred
More informationSection 4.3 Trigonometric Functions of Angles. Unit Circle Trigonometry
Math 0 Section. Section. Trigonometric Functions of Angles Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive
More informationH. PreCalc: REVIEW, Trigonometry
H PreCalc: 7175 REVIEW, Trigonometry Multiple Choice Identify the choice that best completes the statement or answers the question 1 Given and, find and a c, b, d,, 2 What basic trigonometric identity
More information4.7 Inverse Trigonometric Functions
47 Inverse Trigonometric Functions Section 47 Notes Page 1 From our tables in a previous section we know that sin 0 = In inverse trig functions we put in the value and get an angle: 1 We put in an angle
More informationBasics of Cutting Tool Geometry
D. Vikto P. Astakhov, Tool Geomet: Basics Basics of Cutting Tool Geomet Vikto P. Astakhov Fo man eas thee wee diffeent sstems used to define a geat vaiet of angles of faces and edges of cutting tools.
More information2. Right Triangle Trigonometry
2. Right Triangle Trigonometry 2.1 Definition II: Right Triangle Trigonometry 2.2 Calculators and Trigonometric Functions of an Acute Angle 2.3 Solving Right Triangles 2.4 Applications 2.5 Vectors: A Geometric
More informationSection 7.5 Inverse Trigonometric Functions II
Section 7.5 Inverse Trigonometric Functions II Note: A calculator is helpful on some exercises. Bring one to class for this lecture. OBJECTIVE : Evaluating composite Functions involving Inverse Trigonometric
More information