Inverse Circular Functions 6.1 6.2 6.3 6.4 6.1 Inverse Circular Functions Inverse Functions ▪ Inverse Sine Function ▪ Inverse Cosine Function ▪ Inverse Tangent Function ▪ Remaining Inverse Circular Functions ▪ Inverse Function Values Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Restricted Domain and Ranges so that the Inverses of Trigonometric Functions will be Functions On the white board we will discuss the basic sine graph and its inverse …. What restrictions need to be made so that the sin-1 graph is a function? Inverse Sine, Cosecant, and Tangent Functions are restricted to quad I & IV Inverse Cosine, Secant, and Cotangent Functions are restricted to quad I & II Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.1 Example 1 Finding Inverse Sine Values (page 261) Find y in each equation. Find reference angle: Sin of what angle is ½? The angle will be in quadrant IV sin of what angle is ? y = 60˚ = π/3 Ref angle = 30˚ = π/6 y = -30˚ = -π/6 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.1 Example 1 Finding Inverse Sine Values (page 261) Find y in each equation. is not in the domain of the inverse sine function, [–1, 1], so does not exist. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.1 Example 2 Finding Inverse Cosine Values (page 262) Find y in each equation. 0 < y < π cos y = 0 0 < y < π cos y = ½ The cos of what angle is 0? The cos of what angle is ½? y = 90˚ = π/2 y = 60˚ = π/3 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Find the degree measure of θ in each of the following. 6.1 Example 3 Finding Inverse Function Values (Degree-Measured Angles) (page 265) Find the degree measure of θ in each of the following. tan of what angle is ? Reference angle = 45˚ Quad IV y = 60˚ = π/3 y = –45˚ = –π/4 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.1 Ex 4 Finding Inverse Function Values With a Calculator (pg 266) (a) Find y in radians if (b) Find θ in degrees if θ = arccot(–.2528). 0 < θ < 180˚ 0 < y < π cot θ = –.2528 tan θ = –1/.2528 θ = tan-1 (–1/.2528) Ref θ = –75.81286506˚ sec y = – 4 cos y = (–1/4) y = cos-1 (–1/4) y = 1.823476582 θ = 104.1871349° Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Evaluate each expression without a calculator. 6.1 Example 5 Finding Function Values Using Definitions of the Trigonometric Functions (page 266) Evaluate each expression without a calculator. Let θ = sin-1 (2/3) Trying to find cos (θ) = sin-1 restricted to quad I & IV sin-1 is positive (2/3) reference angle will be in quad I Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

cot-1 restricted to quad I & II cot-1 is negative (15/8) 6.1 Example 5(b) Finding Function Values Using Definitions of the Trigonometric Functions (page 266) Let θ = cot-1 (–15/8) Trying to find sec (θ) = cot-1 restricted to quad I & II cot-1 is negative (15/8) reference angle will be in quad II Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.1 Example 6(a) Finding Function Values Using Identities (page 267) Evaluate the expression without a calculator. Let A = arctan 4/3 and B = arccos 12/13 = sin (A – B) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.1 Ex 6(b) Finding Function Values Using Identities (pg 267) Evaluate the expression without a calculator. sin(2 arccot (–5)) Let A = arccot (-5) cot-1 restricted to quad I & II reference angle quad II = sin(2A) = 2 sinA cosA Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.1 Example 7(a) Finding Function Values in Terms of u (page 268) Write , as an algebraic expression in u. Let θ = sec-1u secθ = u cos θ = 1/u Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.1 Example 7(b) Finding Function Values in Terms of u (page 268) Write , u > 0, as an algebraic expression in u. Let θ = cos-1u cos θ = u Quad I Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.1 Example 8 Finding the Optimal Angle of Elevation of a Shot Put (page 269) The optimal angle of elevation θ a shot-putter should aim for to throw the greatest distance depends on the velocity v and the initial height h of the shot. One model for θ that achieves this greatest distance is Suppose a shot-putter can consistently throw the steel ball with h = 7.5 ft and v = 50 ft per sec. At what angle should he throw the ball to maximize distance?

Inverse Circular Functions 6.1 Inverse Circular Functions Inverse Sine, Cosecant, and Tangent Functions are restricted to quad I & IV Inverse Cosine, Secant, and Cotangent Functions are restricted to quad I & II Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Trigonometric Equations I 6.2 Trigonometric Equations I Solving by Linear Methods ▪ Solving by Factoring ▪ Solving by Quadratic Methods ▪ Solving by Using Trigonometric Identities Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.2 Introduction Solve for θ? Quad: Ref. angle:   Quad: Ref. angle: Answer between 0⁰ and 360⁰ Answer between 0 and 2π All answers in degrees All answers in radians I and II 30⁰ 30⁰and 150⁰ π/6 and 5π/6 30⁰+360⁰n, 150⁰+360⁰n where n ε Z π/6 + 2πn, 5π/6 + 2πn where n ε Z Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.2 Introduction Solve for θ? Quad: Ref. angle:   Quad: Ref. angle: Answer between 0⁰ and 360⁰ Answer between 0 and 2π All answers in degrees All answers in radians I and IV 30⁰ 30⁰and 330⁰ π/6 and 11π/6 30⁰+360⁰n, 330⁰+360⁰n where n ε Z π/6 + 2πn, 11π/6 + 2πn where n ε Z Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.2 Introduction Solve for θ? Quad: Ref. angle:   Quad: Ref. angle: Answer between 0⁰ and 360⁰ Answer between 0 and 2π All answers in degrees All answers in radians II and IV 60⁰ 120⁰and 300⁰ 2π/3 and 5π/3 120⁰+360⁰n, 300⁰+360⁰n where n ε Z 2π/3 + 2πn, 5π/3 + 2πn where n ε Z Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

is positive in quadrants I and III. 6.2 Example 1 Solving a Trigonometric Equation by Linear Methods (page 274) is positive in quadrants I and III. The reference angle is 30° Solution set: {30°, 210°} Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.2 Example 2 Solving a Trigonometric Equation by Factoring (page 274) Cot neg in Quad II & IV Ref angle = 45⁰ Where is the x-corr on the axis = 0? or Solution set: {90°, 135°, 270°, 315°} Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.2 Example 3 Solving a Trigonometric Equation by Factoring (page 275) 3u2 – u – 2 = 0 3u2 – 3u + 2u – 2 = 0 3u(u – 1) + 2(u – 1) = 0 (3u + 2)(u – 1) = 0 u = -2/3 or u = 1 sin x = -2/3 will have a ref angle of 0.7297 Answer in quad III & IV π + 0.7297, and 2π – 0.7297 sin x = 1 When is the y-corr 1 on the axis of the unit circle? sin x = -2/3 or sin x = 1 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.2 Example 4 Solving a Trigonometric Equation Using the Quadratic Formula (page 275) Find all solutions of (answer in radians) Use the quadratic formula with a = 1, b = 2, and c = –1 to solve for cos x. Get the program from me. At this time you will be allowed to use Quad Program cos x cos x = 0.41421 or cos x = -2.41421 cos x = 0.41421 will have a reference angle of 1.1437 and the answers will be in quad I and IV and the multiples of 2π … cos x = -2.41421 is NOT possible Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.2 Example 5 Solving a Trigonometric Equation by Squaring (page 276) Since the solution was found by squaring both sides of an equation, we must check that each proposed solution is a solution of the original equation. Not a solution When tan x = √3? Ref angle = 60˚ = π/3 and results will be in quad I and III The possible solutions are Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Solution

6.2 Example 6 Describing a Musical Tone From a Graph (page 277) A basic component of music is a pure tone. The graph below models the sinusoidal pressure y = P in pounds per square foot from a pure tone at time x = t in seconds. The frequency of a pure tone is often measured in hertz. One hertz is equal to one cycle per second and is abbreviated Hz. What is the frequency f in hertz of the pure tone shown in the graph? There are 4 cycles in .0182 seconds. The time for the tone to produce one complete cycle is called the period. Approximate the period T in seconds of the pure tone. The frequency is 220 Hz. Period 2π/440π = 1/220 Four periods cover a time of .0182 seconds. One period = Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.2 Summary Solving Trigonometric Functions: Solving by Linear Methods Solving by Factoring Solving by Quadratic Methods Solving by Using Trigonometric Identities Reminder: All Students Take Calculus Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Trigonometric Equations II 6.3 Trigonometric Equations II Equations with Half-Angles ▪ Equations with Multiple Angles Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.3 Ex 1 Solving an Equation Using a Half-Angle Identity (pg 280) (a) over the interval and (b) give all solutions. π/4 Reference angle Quad I & IV, 45˚= π/4 7π/4 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.3 Ex 2 Solving an Equation With a Double Angle (pg 281) 5π/6 π/6 3π/2 or Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.3 Ex 3 Solving an Equation Using a Multiple Angle Identity (pg 282) Reference angle Quad II & III 60°. Angles 120˚ and 240˚ 2θ = 120˚ + 360˚n & 2θ = 240˚ + 360˚n θ = 60˚ + 180˚n θ = 120˚ + 180˚n θ = 60˚, 240˚, 420˚,… θ = 120˚, 300˚, 480˚,… Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.3 Ex 4 Solving an Equation With a Multiple Angle (page 282) π/2 y/x cot 2x = 0 tan 2x = undefined 3π/2 2x = π/2, 3π/2, 5π/2, 7π/2, 9π/2, 11π/2,… x = π/4, 3π/4, 5π/4, 7π/4, 9π/4, 11π/4, … Because you squared both sides you must check the answers Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.3 Ex 4 Solving an Equation With a Multiple Angle (cont.) ☺ ☺ Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.3 Ex 5 Analyzing Pressures of Upper Harmonics (page 283) Suppose that the E key above middle C is played on a piano. Its fundamental frequency is and its associate pressure is expressed as What are the next four frequencies at which the string will vibrate? Frequency = the reciprocal of the period Period = 2π/b Frequency = b/2π = 660π/2π = 330 Hz f2 = f1 + 330 = 660 Hz f3 = f2 + 330 = 990 Hz f4 = f3 + 330 = 1320 Hz f5 = f4 + 330 = 1650 Hz The string will vibrate at the next four frequencies: 660 Hz, 990 Hz, 1320 Hz, & 1650 Hz Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.3 Summary Solving More Trigonometric Equations: Still need “All Students Take Calculus” If you raise both sides to an even power, you must check your solutions. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Equations Involving Inverse Trigonometric Functions 6.4 Equations Involving Inverse Trigonometric Functions Solving for x in Terms of y Using Inverse Functions ▪ Solving Inverse Trigonometric Equations Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.4 Example 1 Solving an Equation for a Variable Using Inverse Notation (page 287) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.4 Example 3 Solving an Equation Involving Inverse Trigonometric Functions (page 288) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Solve cos-1x = sin-1(3/5) x = cos(sin-1(3/5)) x = 4/5 6.4 add on 5 3 4 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.4 add on Solve Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6.4 add on Solve

6.4 add on Use a Graphing Calculator to Solve Solve (arctan x)4 – 2x + 6 = 0, in the interval [0,6] y1 = (tan-1(x))4 – 2x + 6 y2 = 0 Adjust the window for the domain [0,6] Answer: x ≈ 4.7209 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6.4 Summary Copyright © 2008 Pearson Addison-Wesley. All rights reserved.