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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 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 Equations Graphs of Trigonometric Functions Verte Terminal ra Initial ra Standard position of an angle Figure D. Angles and Degree Measure An angle has three parts: an initial ra, a terminal ra, and a verte (the point of intersection of the two ras), as shown in Figure D.. An angle is in standard position if its initial ra coincides with the positive -ais and its verte is at the origin. We assume that ou are familiar with the degree measure of an angle.* It is common practice to use (the Greek lowercase theta) to represent both an angle and its measure. Angles between 0 and 90 are acute, and angles between 90 and 80 are obtuse. Positive angles are measured counterclockwise, and negative angles are measured clockwise. For instance, Figure D. shows an angle whose measure is. You cannot assign a measure to an angle b simpl knowing where its initial and terminal ras are located. To measure an angle, ou must also know how the terminal ra was revolved. For eample, Figure D. shows that the angle measuring has the same terminal ra as the angle measuring. Such angles are coterminal. In general, if is an angle, then n is a nonzero integer is coterminal with. An angle that is larger than 0 is one whose terminal ra has been revolved more than one full revolution counterclockwise, as shown in Figure D.. You can form an angle whose measure is less than 0 b revolving a terminal ra more than one full revolution clockwise. n0, 0 Coterminal angles Coterminal angles Figure D. Figure D. NOTE It is common to use the smbol to refer to both an angle and its measure. For instance, in Figure D., ou can write the measure of the smaller angle as. *For a more complete review of trigonometr, see Precalculus, th edition, b Larson and Hostetler (Boston, Massachusetts: Houghton Mifflin, 00).

2 D8 APPENDIX D Precalculus Review Radian Measure To assign a radian measure to an angle, consider to be a central angle of a circle of radius, as shown in Figure D.7. The radian measure of is then defined to be the length of the arc of the sector. Because the circumference of a circle is r, the circumference of a unit circle (of radius ) is. This implies that the radian measure of an angle measuring 0 is. In other words, 0 radians. Using radian measure for, the length s of a circular arc of radius r is s r, as shown in Figure D.8. Arc length is s = r. The arc length of the sector is the radian measure of. r = r Unit circle Circle of radius r Figure D.7 Figure D.8 You should know the conversions of the common angles shown in Figure D.9. For other angles, use the fact that 80 is equal to radians. 0 = = 0 = 90 = 80 = 0 = Radian and degree measure for several common angles Figure D.9 EXAMPLE Conversions Between Degrees and Radians a. 0 0 deg 80 deg 9 radians b deg 80 deg radians c. radians rad 80 deg d. radians 9 rad 80 deg 9 rad rad rad 90 rad 80

3 APPENDIX D Precalculus Review D9 Hpotenuse Adjacent Sides of a right triangle Figure D.0 (, ) r r = + Opposite The Trigonometric Functions There are two common approaches to the stud of trigonometr. In one, the trigonometric functions are defined as ratios of two sides of a right triangle. In the other, these functions are defined in terms of a point on the terminal side of an angle in standard position. We define the si trigonometric functions, sine, cosine, tangent, cotangent, secant, and cosecant (abbreviated as sin, cos, etc.), from both viewpoints. Definition of the Si Trigonometric Functions Right triangle definitions, where 0 < < (see Figure D.0). sin opposite cos adjacent tan opposite hpotenuse hpotenuse adjacent csc hpotenuse opposite Circular function definitions, where sec hpotenuse adjacent cot adjacent opposite is an angle (see Figure D.). sin r cos r tan An angle in standard position Figure D. csc r sec r cot The following trigonometric identities are direct consequences of the definitions. is the Greek letter phi. Trigonometric Identities [Note that sin is used to represent sin.] Pthagorean Identities: Reduction Formulas: sin cos tan sec cot csc Sum or Difference of Two Angles: sin cos tan ± ± cos cos sin sin ± Law of Cosines: sin cos ± cos sin tan ± tan tan tan a b c bc cos A b A c a sin sin cos cos tan tan Half Angle Formulas: sin cos cos cos Reciprocal Identities: csc sin sec cos cot tan sin sin cos cos tan tan Double Angle Formulas: sin sin cos cos cos Quotient Identities: tan sin cos cot cos sin sin cos sin

4 D0 APPENDIX D Precalculus Review Evaluating Trigonometric Functions There are two was to evaluate trigonometric functions: () decimal approimations with a calculator (or a table of trigonometric values) and () eact evaluations using trigonometric identities and formulas from geometr. When using a calculator to evaluate a trigonometric function, remember to set the calculator to the appropriate mode degree mode or radian mode. r = 0 = r = (, ) The angle / in standard position Figure D. EXAMPLE Eact Evaluation of Trigonometric Functions Evaluate the sine, cosine, and tangent of. Solution Begin b drawing the angle in standard position, as shown in Figure D.. Then, because 0 radians, ou can draw an equilateral triangle with sides of length and as one of its angles. Because the altitude of this triangle bisects its base, ou know that. Using the Pthagorean Theorem, ou obtain r. Now, knowing the values of,, and r, ou can write the following. sin r cos r tan NOTE All angles in this tet are measured in radians unless stated otherwise. For eample, when we write sin, we mean the sine of radians, and when we write sin, we mean the sine of degrees. 0 Common angles Figure D. 0 The degree and radian measures of several common angles are given in the table below, along with the corresponding values of the sine, cosine, and tangent (see Figure D.). Common First Quadrant Angles Degrees 0 Radians 0 sin cos tan Undefined 90

5 APPENDIX D Precalculus Review D Quadrant II sin : + cos : tan : Quadrant III sin : cos : tan : + Quadrant I sin : + cos : + tan : + Quadrant IV sin : cos : + tan : Quadrant signs for trigonometric functions Figure D. The quadrant signs for the sine, cosine, and tangent functions are shown in Figure D.. To etend the use of the table on the preceding page to angles in quadrants other than the first quadrant, ou can use the concept of a reference angle (see Figure D.), with the appropriate quadrant sign. For instance, the reference angle for is, and because the sine is positive in the second quadrant, ou can write sin sin. Similarl, because the reference angle for 0 is 0, and the tangent is negative in the fourth quadrant, ou can write tan 0 tan 0. Reference angle: Reference angle: Reference angle: Quadrant II = (radians) = 80 (degrees) Figure D. Quadrant III = (radians) = 80 (degrees) Quadrant IV = (radians) = 0 (degrees) EXAMPLE Trigonometric Identities and Calculators Evaluate the trigonometric epression. a. sin b. sec 0 c. cos. Solution a. Using the reduction formula sin sin, ou can write sin sin. b. Using the reciprocal identit sec cos, ou can write sec 0 cos 0. c. Using a calculator, ou can obtain cos. 0.. Remember that. is given in radian measure. Consequentl, our calculator must be set in radian mode.

6 D APPENDIX D Precalculus Review Solving Trigonometric Equations How would ou solve the equation sin 0? You know that is one solution, but this is not the onl solution. An one of the following values of is also a solution....,,,, 0,,,,... You can write this infinite solution set as n: n is an integer. 0 EXAMPLE Solving a Trigonometric Equation = = sin Solution points of sin Figure D. Solve the equation sin. Solution To solve the equation, ou should consider that the sine is negative in Quadrants III and IV and that sin. Thus, ou are seeking values of in the third and fourth quadrants that have a reference angle of. In the interval 0,, the two angles fitting these criteria are and. B adding integer multiples of to each of these solutions, ou obtain the following general solution. n (See Figure D..) or n, where n is an integer. EXAMPLE Solving a Trigonometric Equation Solve cos sin, where 0. Solution Using the double-angle identit cos sin, ou can rewrite the equation as follows. sin sin Given equation Trigonometric identit Quadratic form Factor. If sin 0, then sin and or If sin 0, then sin and Thus, for 0, there are three solutions. cos sin, 0 sin sin 0 sin sin,. or.

7 APPENDIX D Precalculus Review D Graphs of Trigonometric Functions A function f is periodic if there eists a nonzero number p such that f p f for all in the domain of f. The smallest such positive value of p (if it eists) is the period of f. The sine, cosine, secant, and cosecant functions each have a period of, and the other two trigonometric functions have a period of, as shown in Figure D.7. Domain: all reals Range: [, ] Period: = sin Domain: all reals Range: [, ] Period: = cos Domain: all + n Range: (, ) = tan Period: Domain: all n Range: (, ] and [, ) Period: Domain: all + n Range: (, ] and [, ) Period: Domain: all n Range: (, ) Period: = csc = sin = sec = cos = cot = tan The graphs of the si trigonometric functions Figure D.7 Note in Figure D.7 that the maimum value of sin and cos is and the minimum value is. The graphs of the functions a sin b and a cos b oscillate between a and a, and hence have an amplitude of a. Furthermore, because b 0 when 0 and b when b, it follows that the functions a sin b and a cos b each have a period of b. The table below summarizes the amplitudes and periods for some tpes of trigonometric functions. a sin b a tan b a sec b Function Period Amplitude or or or a cos b a cot b a csc b b b b a Not applicable Not applicable

8 D APPENDIX D Precalculus Review f( ) = cos (0, ) Period = Figure D.8 Amplitude = EXAMPLE Sketching the Graph of a Trigonometric Function Sketch the graph of f cos. Solution The graph of f cos has an amplitude of and a period of. Using the basic shape of the graph of the cosine function, sketch one period of the function on the interval 0,, using the following pattern. Maimum: 0, Minimum: Maimum:,, B continuing this pattern, ou can sketch several ccles of the graph, as shown in Figure D.8. Horizontal shifts, vertical shifts, and reflections can be applied to the graphs of trigonometric functions, as illustrated in Eample 7. EXAMPLE 7 Shifts of Graphs of Trigonometric Functions Sketch the graphs of the following functions. a. f sin b. f sin c. f sin Solution a. To sketch the graph of f sin, shift the graph of sin to the left units, as shown in Figure D.9(a). b. To sketch the graph of f sin, shift the graph of sin up two units, as shown in Figure D.9(b). c. To sketch the graph of f sin, shift the graph of sin up two units and to the right units, as shown in Figure D.9(c). f ( ) = sin + ( ( f ( ) = + sin f( ) = + sin ( ( = sin = sin = sin (a) Horizontal shift to the left (b) Vertical shift upward (c) Horizontal and vertical shifts Transformations of the graph of sin Figure D.9

9 APPENDIX D Precalculus Review D E X E R C I S E S F O R APPENDIX D. In Eercises and, determine two coterminal angles (one positive and one negative) for each given angle. Epress our answers in degrees.. (a) (b) =. (a) = 00 (b) = 0 = 0 0. Angular Speed A car is moving at the rate of 0 miles per hour, and the diameter of its wheels is. feet. (a) Find the number of revolutions per minute that the wheels are rotating. (b) Find the angular speed of the wheels in radians per minute. In Eercises and, determine all si trigonometric functions for the angle.. (a) (b) (, ) In Eercises and, determine two coterminal angles (one positive and one negative) for each given angle. Epress our answers in radians.. (a) (b) = 9 =. (a) (b) (, ). (a) (b) = 9 In Eercises and, epress the angles in radian measure as multiples of and as decimals accurate to three decimal places.. (a) 0 (b) 0 (c) (d) 0. (a) 0 (b) 0 (c) 70 (d) = 8 9 In Eercises and, determine the quadrant in which. (a) sin < 0 and cos < 0 (b) sec > 0 and cot < 0. (a) sin > 0 and cos < 0 (b) csc < 0 and tan > 0 In Eercises 8, evaluate the trigonometric function.. sin. sin cos (8, ) tan (, ) lies. In Eercises 7 and 8, epress the angles in degree measure. 7. (a) (b) (c) 7 (d) (a) (b) (c) (d) Let r represent the radius of a circle, the central angle (measured in radians), and s the length of the arc subtended b the angle. Use the relationship s r to complete the table. r 8 ft in. 8 cm 7. cos 8. cot sec csc s ft 9 in. 8 mi.

10 D APPENDIX D Precalculus Review In Eercises 9, evaluate the sine, cosine, and tangent of each angle without using a calculator. 9. (a) 0 0. (a) 0 (b) 0 (b) 0 (c) (c) (d) (d). (a). (a) 70 (b) (b) 0 (c) (d) In Eercises, use a calculator to evaluate the trigonometric functions to four significant digits. (c) (d). (a) sin 0. (a) sec (b) csc 0 (b) sec. (a) tan. (a) cot. 9 (b) tan 0 (b) tan Airplane Ascent An airplane leaves the runwa climbing at 8 with a speed of 7 feet per second (see figure). Find the altitude a of the plane after minute Height of a Mountain In traveling across flat land, ou notice a mountain directl in front of ou. Its angle of elevation (to the peak) is.. After ou drive miles closer to the mountain, the angle of elevation is 9. Approimate the height of the mountain.. 9 mi (not to scale) In Eercises, determine the period and amplitude of each function.. (a) sin (b) sin a In Eercises 7 0, find two solutions of each equation. Epress the results in radians 0 <. Do not use a calculator. 7. (a) cos 8. (a) sec (b) cos (b) sec 9. (a) tan 0. (a) (b) cot (b) sin sin. (a) (b) cos sin In Eercises 8, solve the equation for. sin. tan. tan tan 0. cos cos. sec csc csc. sin cos 7. cos sin 8. cos cos 0 <.. sin. cos 0

11 APPENDIX D Precalculus Review D7 In Eercises 8, find the period of the function.. tan. 7 tan 7. sec 8. csc Writing In Eercises 9 and 0, use a graphing utilit to graph each function f on the same set of coordinate aes for c, c, c, and c. Give a written description of the change in the graph caused b changing c. 9. (a) f c sin 0. (a) f sin c (b) f cosc (b) f sin c (c) f cos c (c) f c cos In Eercises, sketch the graph of the function.. sin. cos 7. Sales Sales S, in thousands of units, of a seasonal product are modeled b t S 8.. cos where t is the time in months (with t corresponding to Januar and t corresponding to December). Use a graphing utilit to graph the model for S and determine the months when sales eceed 7,000 units. 8. Investigation Two trigonometric functions f and g have a period of, and their graphs intersect at.. (a) Give one smaller and one larger positive value of where the functions have the same value. (b) Determine one negative value of where the graphs intersect. (c) Is it true that f. g.? Give a reason for our answer.. sin.. csc. 7. sec sin 0.. cos. sin tan tan csc cos Pattern Recognition In Eercises 9 and 70, use a graphing utilit to compare the graph of f with the given graph. Tr to improve the approimation b adding a term to f. Use a graphing utilit to verif that our new approimation is better than the original. Can ou find other terms to add to make the approimation even better? What is the pattern? (In Eercise 9, sine terms can be used to improve the approimation and in Eercise 70, cosine terms can be used.) 9. f sin sin Graphical Reasoning In Eercises and, find a, b, and c such that the graph of the function matches the graph in the figure.. a cosb c. a sinb c 70. f cos 9 cos. Think About It Sketch the graphs of f sin, g sin and In general, how are the graphs of, h sin f and f. related to the graph of f?. Think About It The model for the height h of a Ferris wheel car is h 0 sin 8t where t is measured in minutes. (The Ferris wheel has a radius of 0 feet.) This model ields a height of feet when t 0. Alter the model so that the height of the car is foot when t 0.

4.1 Radian and Degree Measure

4.1 Radian and Degree Measure Date: 4.1 Radian and Degree Measure Syllabus Objective: 3.1 The student will solve problems using the unit circle. Trigonometry means the measure of triangles. Terminal side Initial side Standard Position

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

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

10.2 The Unit Circle: Cosine and Sine

10.2 The Unit Circle: Cosine and Sine 0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on

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

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

Triangle Definition of sin and cos

Triangle Definition of sin and cos Triangle Definition of sin and cos Then Consider the triangle ABC below. Let A be called. A HYP (hpotenuse) ADJ (side adjacent to the angle ) B C OPP (side opposite to the angle ) sin OPP HYP BC AB ADJ

More information

5.2 Unit Circle: Sine and Cosine Functions

5.2 Unit Circle: Sine and Cosine Functions Chapter 5 Trigonometric Functions 75 5. Unit Circle: Sine and Cosine Functions In this section, you will: Learning Objectives 5..1 Find function values for the sine and cosine of 0 or π 6, 45 or π 4 and

More information

Definition III: Circular Functions

Definition III: Circular Functions SECTION 3.3 Definition III: Circular Functions Copyright Cengage Learning. All rights reserved. Learning Objectives 1 2 3 4 Evaluate a trigonometric function using the unit circle. Find the value of a

More information

Inverse Trigonometric Functions. Inverse Sine Function 4.71 FIGURE. Definition of Inverse Sine Function. The inverse sine function is defined by

Inverse Trigonometric Functions. Inverse Sine Function 4.71 FIGURE. Definition of Inverse Sine Function. The inverse sine function is defined by 0_007.qd /7/05 :0 AM Page Section.7.7 Inverse Trigonometric Functions Inverse Trigonometric Functions What ou should learn Evaluate and graph the inverse sine function. Evaluate and graph the other inverse

More information

Angles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry

Angles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible

More information

MA Lesson 19 Summer 2016 Angles and Trigonometric Functions

MA Lesson 19 Summer 2016 Angles and Trigonometric Functions DEFINITIONS: An angle is defined as the set of points determined by two rays, or half-lines, 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 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

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

MPE Review Section II: Trigonometry

MPE Review Section II: Trigonometry MPE Review Section II: Trigonometry Review similar triangles, right triangles, and the definition of the sine, cosine and tangent functions of angles of a right triangle In particular, recall that the

More information

Trigonometric Functions and Triangles

Trigonometric Functions and Triangles Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between

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

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

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

Math 115 Spring 2014 Written Homework 10-SOLUTIONS Due Friday, April 25

Math 115 Spring 2014 Written Homework 10-SOLUTIONS Due Friday, April 25 Math 115 Spring 014 Written Homework 10-SOLUTIONS Due Friday, April 5 1. Use the following graph of y = g(x to answer the questions below (this is NOT the graph of a rational function: (a State the domain

More information

FLC Ch 1 & 3.1. A ray AB, denoted, is the union of and all points on such that is between and. The endpoint of the ray AB is A.

FLC Ch 1 & 3.1. A ray AB, denoted, is the union of and all points on such that is between and. The endpoint of the ray AB is A. Math 335 Trigonometry Sec 1.1: Angles Definitions A line is an infinite set of points where between any two points, there is another point on the line that lies between them. Line AB, A line segment is

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

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

Trigonometry (Chapters 4 5) Sample Test #1 First, a couple of things to help out:

Trigonometry (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 information

Trigonometry LESSON ONE - Degrees and Radians Lesson Notes

Trigonometry LESSON ONE - Degrees and Radians Lesson Notes 210 180 = 7 6 Trigonometry Example 1 Define each term or phrase and draw a sample angle. Angle Definitions a) angle in standard position: Draw a standard position angle,. b) positive and negative angles:

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

Chapter 6: Periodic Functions

Chapter 6: Periodic Functions Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a triangle, and related to points on a circle. We noticed how the x and y values

More information

Section 6-3 Double-Angle and Half-Angle Identities

Section 6-3 Double-Angle and Half-Angle Identities 6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities

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

Trigonometry Notes Sarah Brewer Alabama School of Math and Science. Last Updated: 25 November 2011

Trigonometry Notes Sarah Brewer Alabama School of Math and Science. Last Updated: 25 November 2011 Trigonometry Notes Sarah Brewer Alabama School of Math and Science Last Updated: 25 November 2011 6 Basic Trig Functions Defined as ratios of sides of a right triangle in relation to one of the acute angles

More information

Chapter 6 Trigonometric Functions of Angles

Chapter 6 Trigonometric Functions of Angles 6.1 Angle Measure Chapter 6 Trigonometric Functions of Angles In Chapter 5, we looked at trig functions in terms of real numbers t, as determined by the coordinates of the terminal point on the unit circle.

More information

Trigonometric. Functions CHAPTER OUTLINE

Trigonometric. Functions CHAPTER OUTLINE bar5969_ch05_463-478.qd 0/5/008 06:53 PM Page 463 pinnacle 0:MHIA065:SE:CH 05: CHAPTER 5 Trigonometric Functions C TRIGONOMETRIC functions seem to have had their origins with the Greeks investigation of

More information

CRASH COURSE IN PRECALCULUS

CRASH COURSE IN PRECALCULUS CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 2012, Brooks/Cole

More information

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its

More information

Trigonometry Chapter 3 Lecture Notes

Trigonometry Chapter 3 Lecture Notes Ch Notes Morrison Trigonometry Chapter Lecture Notes Section. Radian Measure I. Radian Measure A. Terminology When a central angle (θ) intercepts the circumference of a circle, the length of the piece

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

ACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude

ACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height

More information

Solutions to Exercises, Section 5.1

Solutions to Exercises, Section 5.1 Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

More information

TRIGONOMETRIC FUNCTIONS

TRIGONOMETRIC FUNCTIONS Chapter TRIGONOMETRIC FUNCTIONS.1 Introduction A mathematician knows how to solve a problem, he can not solve it. MILNE The word trigonometr is derived from the Greek words trigon and metron and it means

More information

RIGHT TRIANGLE TRIGONOMETRY

RIGHT 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 information

Mid-Chapter Quiz: Lessons 4-1 through 4-4

Mid-Chapter Quiz: Lessons 4-1 through 4-4 Find the exact values of the six trigonometric functions of θ. Find the value of x. Round to the nearest tenth if necessary. 1. The length of the side opposite is 24, the length of the side adjacent to

More information

Section 3.1 Radian Measure

Section 3.1 Radian Measure Section.1 Radian Measure Another way of measuring angles is with radians. This allows us to write the trigonometric functions as functions of a real number, not just degrees. A central angle is an angle

More information

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b

More information

29 Wyner PreCalculus Fall 2016

29 Wyner PreCalculus Fall 2016 9 Wyner PreCalculus Fall 016 CHAPTER THREE: TRIGONOMETRIC EQUATIONS Review November 8 Test November 17 Trigonometric equations can be solved graphically or algebraically. Solving algebraically involves

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

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

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

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

Higher. Functions and Graphs. Functions and Graphs 18

Higher. Functions and Graphs. Functions and Graphs 18 hsn.uk.net Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 8 Sets 8 Functions 9 Composite Functions 4 Inverse Functions 5 Eponential Functions 4 6 Introduction to Logarithms

More information

10.6 The Inverse Trigonometric Functions

10.6 The Inverse Trigonometric Functions 0.6 The Inverse Trigonometric Functions 70 0.6 The Inverse Trigonometric Functions As the title indicates, in this section we concern ourselves with finding inverses of the (circular) trigonometric functions.

More information

Similar Right Triangles

Similar Right Triangles 9.1 Similar Right Triangles Goals p Solve problems involving similar right triangles formed b the altitude drawn to the hpotenuse of a right triangle. p Use a geometric mean to solve problems. THEOREM

More information

Section 10-5 Parametric Equations

Section 10-5 Parametric Equations 88 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. A hperbola with the following graph: (2, ) (0, 2) 6. A hperbola with the following graph: (, ) (2, 2) C In Problems 7 2, find the coordinates of an foci relative

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 110 B) 120 C) 60 D) 150

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 110 B) 120 C) 60 D) 150 Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Convert the angle to decimal degrees and round to the nearest hundredth of a degree. ) 56

More information

1) Convert 13 32' 47" to decimal degrees. Round your answer to four decimal places.

1) Convert 13 32' 47 to decimal degrees. Round your answer to four decimal places. PRECLCULUS FINL EXM, PRCTICE UNIT ONE TRIGONOMETRIC FUNCTIONS ) Convert ' 47" to decimal degrees. Round your answer to four decimal places. ) Convert 5.6875 to degrees, minutes, and seconds. Round to the

More information

The Quadratic Function

The Quadratic Function 0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral

More information

Trigonometric Identities and Equations

Trigonometric Identities and Equations LIALMC07_0768.QXP /6/0 0:7 AM Page 605 7 Trigonometric Identities and Equations In 8 Michael Faraday discovered that when a wire passes by a magnet, a small electric current is produced in the wire. Now

More information

Give an expression that generates all angles coterminal with the given angle. Let n represent any integer. 9) 179

Give an expression that generates all angles coterminal with the given angle. Let n represent any integer. 9) 179 Trigonometry Chapters 1 & 2 Test 1 Name Provide an appropriate response. 1) Find the supplement of an angle whose measure is 7. Find the measure of each angle in the problem. 2) Perform the calculation.

More information

y = a sin ωt or y = a cos ωt then the object is said to be in simple harmonic motion. In this case, Amplitude = a (maximum displacement)

y = a sin ωt or y = a cos ωt then the object is said to be in simple harmonic motion. In this case, Amplitude = a (maximum displacement) 5.5 Modelling Harmonic Motion Periodic behaviour happens a lot in nature. Examples of things that oscillate periodically are daytime temperature, the position of a weight on a spring, and tide level. If

More information

Mathematical Procedures

Mathematical Procedures CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,

More information

Chapter 8A - Angles and Circles

Chapter 8A - Angles and Circles - Chapter 8A Chapter 8A - Angles and Circles Man applications of calculus use trigonometr, which is the stud of angles and functions of angles and their application to circles, polgons, and science. We

More information

Section 7.5 Unit Circle Approach; Properties of the Trigonometric Functions. cosθ a. = cosθ a.

Section 7.5 Unit Circle Approach; Properties of the Trigonometric Functions. cosθ a. = cosθ a. Section 7.5 Unit Circle Approach; Properties of the Trigonometric Functions A unit circle is a circle with radius = 1 whose center is at the origin. Since we know that the formula for the circumference

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

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

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the

More information

1. Introduction sine, cosine, tangent, cotangent, secant, and cosecant periodic

1. Introduction sine, cosine, tangent, cotangent, secant, and cosecant periodic 1. Introduction There are six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant; abbreviated as sin, cos, tan, cot, sec, and csc respectively. These are functions of a single

More information

About Trigonometry. Triangles

About Trigonometry. Triangles About Trigonometry TABLE OF CONTENTS About Trigonometry... 1 What is TRIGONOMETRY?... 1 Triangles... 1 Background... 1 Trigonometry with Triangles... 1 Circles... 2 Trigonometry with Circles... 2 Rules/Conversion...

More information

Solution Guide for Chapter 6: The Geometry of Right Triangles

Solution Guide for Chapter 6: The Geometry of Right Triangles Solution Guide for Chapter 6: The Geometry of Right Triangles 6. THE THEOREM OF PYTHAGORAS E-. Another demonstration: (a) Each triangle has area ( ). ab, so the sum of the areas of the triangles is 4 ab

More information

ANALYTICAL METHODS FOR ENGINEERS

ANALYTICAL METHODS FOR ENGINEERS UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

More information

9.1 Trigonometric Identities

9.1 Trigonometric Identities 9.1 Trigonometric Identities r y x θ x -y -θ r sin (θ) = y and sin (-θ) = -y r r so, sin (-θ) = - sin (θ) and cos (θ) = x and cos (-θ) = x r r so, cos (-θ) = cos (θ) And, Tan (-θ) = sin (-θ) = - sin (θ)

More information

Section 8.1: The Inverse Sine, Cosine, and Tangent Functions

Section 8.1: The Inverse Sine, Cosine, and Tangent Functions Section 8.1: The Inverse Sine, Cosine, and Tangent Functions The function y = sin x doesn t pass the horizontal line test, so it doesn t have an inverse for every real number. But if we restrict the function

More information

Section V.2: Magnitudes, Directions, and Components of Vectors

Section V.2: Magnitudes, Directions, and Components of Vectors Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions

More information

SUM AND DIFFERENCE FORMULAS

SUM AND DIFFERENCE FORMULAS SUM AND DIFFERENCE FORMULAS Introduction We have several identities that we are concentrating on in this section: o Difference Identities for Cosine o Sum Identities for Cosine o Cofunction Identities

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

Chapter 5 The Trigonometric Functions

Chapter 5 The Trigonometric Functions P a g e 40 Chapter 5 The Trigonometric Functions Section 5.1 Angles Initial side Terminal side Standard position of an angle Positive angle Negative angle Coterminal Angles Acute angle Obtuse angle Complementary

More information

SAT Subject Math Level 2 Facts & Formulas

SAT Subject Math Level 2 Facts & Formulas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses

More information

Inverse Trigonometric Functions

Inverse Trigonometric Functions Inverse Trigonometric Functions I. Four Facts About Functions and Their Inverse Functions:. A function must be one-to-one (an horizontal line intersects it at most once) in order to have an inverse function..

More information

Unit 6 Trigonometric Identities, Equations, and Applications

Unit 6 Trigonometric Identities, Equations, and Applications Accelerated Mathematics III Frameworks Student Edition Unit 6 Trigonometric Identities, Equations, and Applications nd Edition Unit 6: Page of 3 Table of Contents Introduction:... 3 Discovering the Pythagorean

More information

y = rsin! (opp) x = z cos! (adj) sin! = y z = The Other Trig Functions

y = 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 information

Trigonometric functions

Trigonometric functions Chapter0 Trigonometric functions Sllabus reference: 3. Contents: A B C D E F Periodic behaviour The sine function Modelling using sine functions The cosine function The tangent function General trigonometric

More information

Chapter 5: Trigonometric Functions of Angles

Chapter 5: Trigonometric Functions of Angles Chapter 5: Trigonometric Functions of Angles In the previous chapters we have explored a variety of functions which could be combined to form a variety of shapes. In this discussion, one common shape has

More information

MAC 1114: Trigonometry Notes

MAC 1114: Trigonometry Notes MAC 1114: Trigonometry Notes Instructor: Brooke Quinlan Hillsborough Community College Section 7.1 Angles and Their Measure Greek Letters Commonly Used in Trigonometry Quadrant II Quadrant III Quadrant

More information

Finding trig functions given another or a point (i.e. sin θ = 3 5. Finding trig functions given quadrant and line equation (Problems in 6.

Finding trig functions given another or a point (i.e. sin θ = 3 5. Finding trig functions given quadrant and line equation (Problems in 6. 1 Math 3 Final Review Guide This is a summary list of many concepts covered in the different sections and some examples of types of problems that may appear on the Final Exam. The list is not exhaustive,

More information

Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123

Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123 Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from

More information

Lesson 6.1 Exercises, pages

Lesson 6.1 Exercises, pages Lesson 6. Eercises, pages 7 80 A. Use technolog to determine the value of each trigonometric ratio to the nearest thousandth. a) sin b) cos ( 6 ) c) cot 7 d) csc 8 0.89 0. tan 7 sin 8 0..0. Sketch each

More information

15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors

15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential

More information

Inverse Circular Function and Trigonometric Equation

Inverse Circular Function and Trigonometric Equation Inverse Circular Function and Trigonometric Equation 1 2 Caution The 1 in f 1 is not an exponent. 3 Inverse Sine Function 4 Inverse Cosine Function 5 Inverse Tangent Function 6 Domain and Range of Inverse

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

6.1 - Introduction to Periodic Functions

6.1 - Introduction to Periodic Functions 6.1 - Introduction to Periodic Functions Periodic Functions: Period, Midline, and Amplitude In general: A function f is periodic if its values repeat at regular intervals. Graphically, this means that

More information

TRIGONOMETRIC FUNCTIONS

TRIGONOMETRIC FUNCTIONS 0 YEAR The Improving Mathematics Education in Schools (TIMES) Project TRIGNMETRIC FUNCTINS MEASUREMENT AND GEMETRY Module 25 A guide for teachers - Year 0 June 20 Trigonometric Functions (Measurement and

More information

Self-Paced Study Guide in Trigonometry. March 31, 2011

Self-Paced Study Guide in Trigonometry. March 31, 2011 Self-Paced Study Guide in Trigonometry March 1, 011 1 CONTENTS TRIGONOMETRY Contents 1 How to Use the Self-Paced Review Module Trigonometry Self-Paced Review Module 4.1 Right Triangles..........................

More information

Analyzing the Graph of a Function

Analyzing the Graph of a Function SECTION A Summar of Curve Sketching 09 0 00 Section 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure 5 A Summar of Curve Sketching Analze and sketch the graph of a function Analzing the

More information

Graphing Trigonometric Skills

Graphing Trigonometric Skills Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE

More information

In this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)

In this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1) Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and

More information

The Circular Functions and Their Graphs

The Circular Functions and Their Graphs LIALMC_78.QXP // : AM Page 5 The Circular Functions and Their Graphs In August, the planet Mars passed closer to Earth than it had in almost, ears. Like Earth, Mars rotates on its ais and thus has das

More information

6.3 Polar Coordinates

6.3 Polar Coordinates 6 Polar Coordinates Section 6 Notes Page 1 In this section we will learn a new coordinate sstem In this sstem we plot a point in the form r, As shown in the picture below ou first draw angle in standard

More information

Final Exam Practice--Problems

Final Exam Practice--Problems Name: Class: Date: ID: A Final Exam Practice--Problems Problem 1. Consider the function f(x) = 2( x 1) 2 3. a) Determine the equation of each function. i) f(x) ii) f( x) iii) f( x) b) Graph all four functions

More information

Trigonometric Functions

Trigonometric Functions Trigonometric Functions MATH 10, Precalculus J. Robert Buchanan Department of Mathematics Fall 011 Objectives In this lesson we will learn to: identify a unit circle and describe its relationship to real

More information

Trigonometric Identities and Conditional Equations C

Trigonometric Identities and Conditional Equations C Trigonometric Identities and Conditional Equations C TRIGONOMETRIC functions are widely used in solving real-world problems and in the development of mathematics. Whatever their use, it is often of value

More information

Trigonometry. Week 1 Right Triangle Trigonometry

Trigonometry. Week 1 Right Triangle Trigonometry Trigonometry Introduction Trigonometry is the study of triangle measurement, but it has expanded far beyond that. It is not an independent subject of mathematics. In fact, it depends on your knowledge

More information

Years t. Definition Anyone who has drawn a circle using a compass will not be surprised by the following definition of the circle: x 2 y 2 r 2 304

Years t. Definition Anyone who has drawn a circle using a compass will not be surprised by the following definition of the circle: x 2 y 2 r 2 304 Section The Circle 65 Dollars Purchase price P Book value = f(t) Salvage value S Useful life L Years t FIGURE 3 Straight-line depreciation. The Circle Definition Anone who has drawn a circle using a compass

More information