7 Trigonometric Identities and Equations © 2008 Pearson Addison-Wesley. All rights reserved Sections 7.5–7.7

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Inverse Circular Functions 7.6Trigonometric Equations 7.7Equations Involving Inverse Trigonometric Functions Trigonometric Identities and Equations 7

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Inverse Circular Functions 7.5 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 Find y in each equation. 7.5 Example 1 Finding Inverse Sine Values (page 688)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 1 Finding Inverse Sine Values (cont.)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 1 Finding Inverse Sine Values (cont.) is not in the domain of the inverse sine function, [–1, 1], so does not exist. A graphing calculator will give an error message for this input.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Find y in each equation. 7.5 Example 2 Finding Inverse Cosine Values (page 689)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Find y in each equation. 7.5 Example 2 Finding Inverse Cosine Values (page 689)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 3 Finding Inverse Function Values (Degree- Measured Angles) (page 692) Find the degree measure of θ in each of the following.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4 Finding Inverse Function Values With a Calculator (page 693) (a)Find y in radians if With the calculator in radian mode, enter as y =

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4(b) Finding Inverse Function Values With a Calculator (page 693) (b)Find θ in degrees if θ = arccot(–.2528). A calculator gives the inverse cotangent value of a negative number as a quadrant IV angle. The restriction on the range of arccotangent implies that the angle must be in quadrant II, so, with the calculator in degree mode, enter arccot(–.2528) as

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4(b) Finding Inverse Function Values With a Calculator (cont.) θ = °

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 5 Finding Function Values Using Definitions of the Trigonometric Functions (page 693) Evaluate each expression without a calculator. Since arcsin is defined only in quadrants I and IV, and is positive, θ is in quadrant I.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 5(a) Finding Function Values Using Definitions of the Trigonometric Functions (cont.)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 5(b) Finding Function Values Using Definitions of the Trigonometric Functions (page 693) Since arccot is defined only in quadrants I and II, and is negative, θ is in quadrant II.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 5(b) Finding Function Values Using Definitions of the Trigonometric Functions (cont.)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 6(a) Finding Function Values Using Identities (page 694) Evaluate the expression without a calculator. Use the cosine difference identity:

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 6(a) Finding Function Values Using Identities (cont.) Sketch both A and B in quadrant I. Use the Pythagorean theorem to find the missing side.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 6(a) Finding Function Values Using Identities (cont.)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 6(b) Finding Function Values Using Identities (page 694) Evaluate the expression without a calculator. Use the double-angle sine identity: sin(2 arccot (–5)) Let A = arccot (–5), so cot A = –5.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 6(b) Finding Function Values Using Identities (cont.) Sketch A in quadrant II. Use the Pythagorean theorem to find the missing side.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 6(b) Finding Function Values Using Identities (cont.)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 7(a) Finding Function Values in Terms of u (page 695) Write, as an algebraic expression in u. Sketch θ in quadrant I. Use the Pythagorean theorem to find the missing side.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 7(a) Finding Function Values in Terms of u (cont.)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 7(b) Finding Function Values in Terms of u (page 695) Write, u > 0, as an algebraic expression in u. Sketch θ in quadrant I. Use the Pythagorean theorem to find the missing side.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 7(b) Finding Function Values in Terms of u (cont.) Use the double-angle sine identity to find sin 2θ.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 8 Finding the Optimal Angle of Elevation of a Shot Put (page 696) 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.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 8 Finding the Optimal Angle of Elevation of a Shot Put (cont.) 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? One model for θ that achieves this greatest distance is

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Trigonometric Equations 7.6 Solving by Linear Methods ▪ Solving by Factoring ▪ Solving by Quadratic Methods ▪ Solving by Using Trigonometric Identities ▪ Equations with Half-Angles ▪ Equations with Multiple Angles ▪ Applications

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 1 Solving a Trigonometric Equation by Linear Methods (page 701) is positive in quadrants I and III. The reference angle is 30° because

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 1 Solving a Trigonometric Equation by Linear Methods (cont.) Solution set: {30°, 210°}

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 2 Solving a Trigonometric Equation by Factoring (page 701) or Solution set: {90°, 135°, 270°, 315°}

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 3 Solving a Trigonometric Equation by Factoring (page 702)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 3 Solving a Trigonometric Equation by Factoring (cont.) has one solution, has two solutions, the angles in quadrants III and IV with the reference angle : and

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4 Solving a Trigonometric Equation Using the Quadratic Formula (page 702) Find all solutions of Use the quadratic formula with a = 1, b = 2, and c = –1 to solve for cos x.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4 Solving a Trigonometric Equation Using the Quadratic Formula (cont.) Since there are two solutions, one in quadrant I and the other in quadrant IV. Since, there are no solutions for this value of cos x. To find all solutions, add integer multiples of the period of cosine, 2

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 5 Solving a Trigonometric Equation by Squaring (page 703) Square both sides. The possible solutions are

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 5 Solving a Trigonometric Equation by Squaring (cont.) 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 solutionSolution

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 6 Solving an Equation Using a Half-Angle Identity (page 704) (a)over the interval and (b)give all solutions. is not in the requested domain.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 6 Solving an Equation Using a Half-Angle Identity (cont.) This is a cosine curve with period

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 7 Solving an Equation With a Double Angle (page 705) Factor. or

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 8 Solving an Equation Using a Multiple Angle Identity (page 705) From the given interval 0 ° ≤ θ < 360°, the interval for 2θ is 0 ° ≤ 2θ < 720°.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 8 Solving an Equation Using a Multiple Angle Identity (cont.) Since cosine is negative in quadrants II and III, solutions over this interval are

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 9 Describing a Musical Tone From a Graph (page 706) 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.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 9(a) Describing a Musical Tone From a Graph (page 706) 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 frequency is 220 Hz.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 9(b) Describing a Musical Tone From a Graph (page 706) The time for the tone to produce one complete cycle is called the period. Approximate the period T in seconds of the pure tone. Four periods cover a time of.0182 seconds. One period =

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 9(c) Describing a Musical Tone From a Graph (page 706) Use a calculator to estimate the first solution to the equation that makes y =.002 over the interval [0,.0182]. The first point of intersection is at about x = sec.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 10 Analyzing Pressures of Upper Harmonics (page 707) Suppose that the E key above middle C is played on a piano. Its fundamental frequency is and its associate pressure is expressed as (a)What are the next four frequencies at which the string will vibrate?

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 10 Analyzing Pressures of Upper Harmonics (cont.) (b)What are the pressures corresponding to these four upper harmonics?

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equations Involving Inverse Trigonometric Functions 7.7 Solving for x in Terms of y Using Inverse Functions ▪ Solving Inverse Trigonometric Equations

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 1 Solving an Equation for a Variable Using Inverse Notation (page 713)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 2 Solving an Equation Involving an Inverse Trigonometric Function (page 713)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 3 Solving an Equation Involving Inverse Trigonometric Functions (page 714)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 3 Solving an Equation Involving Inverse Trigonometric Functions (cont.) Sketch u in quadrant I. Use the Pythagorean theorem to find the missing side.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4 Solving an Inverse Trigonometric Equation Using an Identity (page 714) Isolate one inverse function on one side of the equation: Sine difference identity By definition, the arcsine function is defined in quadrants I and IV, so

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4 Solving an Inverse Trigonometric Equation Using an Identity (cont.) From equation (1), Sketch u in Quadrant III. Use the Pythagorean theorem to find the missing side. By definition, the range of arccos x is so the intersection of the two ranges is

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4 Solving an Inverse Trigonometric Equation Using an Identity (cont.) Substitute into equation (2): Square both sides.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4 Solving an Inverse Trigonometric Equation Using an Identity (cont.) Check each potential solution. There is no value of x in the given domain such that

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4 Solving an Inverse Trigonometric Equation Using an Identity (cont.) Range of arcsine is

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4 Solving an Inverse Trigonometric Equation Using an Identity (cont.)