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NAME DATE 5-3 PERIOD Study Guide and Intervention Solving Trigonometric Equations Use Algebraic Techniques to Solve To solve a trigonometric equation, you may need to apply algebraic methods. These methods include isolating the trigonometric expression, taking the square root of each side, factoring and applying the Zero-Product Property, applying the quadratic formula, or rewriting using a single trigonometric function. In this lesson, we will consider conditional trigonometric equations, or equations that may be true for certain values of the variable but false for others. Example 1 Find all solutions of tan x cos x - cos x = 0 on the interval [0, 2π). tan x cos x - cos x = 0 Original equation cos x (tan x - 1) = 0 Factor. cos x = 0 or tan x - 1 = 0 Set each factor equal to 0. 3π π x=− or − 2 tan x = 1 2 5π π x=− or − 4 4 3π π or − , tan x is undefined, so the solutions of the original When x = − 2 2 5π π or − . When you solve for all values of x, the solution should equation are − 4 4 be represented as x + 2nπ for sin x and cos x and x + nπ for tan x, where n 5π π + nπ or − + nπ. is any integer. The solutions are − 4 Example 2 4 Find all solutions of sin x + √ 3 = -sin x. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 3 = -sin x sin x + √ √ 2 sin x + 3 = 0 2 sin x = - √ 3 Original equation Add sin x to each side. Subtract √ 3 from each side. √3 Divide each side by 2. 2 5π 4π x=− or − Solve for x. 3 3 5π 4π The solutions are − + 2nπ or − + 2nπ. 3 3 sin x = - − Exercises Solve each equation for all values of x. 1. cos x = -1 2. sin3 x - 4 sin x = 0 3. sin x cos x -3 cos x = 0 4. 2 sin3 x = sin x Find all solutions of each equation on the interval [0, 2π). 5. 2 cos x = 1 6. 5 + 2 sin x - 7 = 0 7. 4 sin2 x tan x = tan x 8. 2 cos x - √ 3 =0 Chapter 5 16 Glencoe Precalculus NAME 5-3 DATE Study Guide and Intervention PERIOD (continued) Solving Trigonometric Equations Use Trigonometric Identities to Solve You can use trigonometric identities along with algebraic methods to solve trigonometric equations. Be careful to check all solutions in the original equation to make sure they are valid solutions. Example 1 Find all solutions of 2 tan2 x - sec2 x + 3 = 1 - 2 tan x on the interval [0, 2π). 2 tan2 x - sec2 x + 3 = 1 - 2 tan x Original equation 2 2 2 tan x - (tan x + 1) + 3 = 1 - 2 tan x sec2 x = tan2 x + 1 tan2 x + 2 = 1 - 2 tan x Simplify. 2 tan x + 2 tan x + 1 = 0 Simplify. 2 (tan x + 1) = 0 Factor. tan x = -1 Take the square root of each side. 3π 7π x = − or − Solve for x on [0, 2π). 4 Example 2 Find all solutions of 1 + cos x = sin x on the interval [0, 2π). 1 + cos x = sin x Original equation 2 2 (1 + cos x) = (sin x) Square each side. 2 2 Multiply. 1 + 2 cos x + cos x = sin x 2 2 1 + 2 cos x + cos x = 1 - cos x Pythagorean Identity 2 2 cos x + 2 cos x = 0 Simplify. 2 cos x (cos x + 1) = 0 Factor. cos x = 0 or cos x = -1 Zero Product Property 3π π x = −, π, − Solve for x on [0, 2π). 2 Lesson 5-3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 4 2 Exercises Solve each equation for all values of x. 1. tan2 x = 1 2. 2 sin2 x - cos x = 1 3. sin x cos x -3 cos x = 0 4. cos2 x + sin x + 1 = 0 Find all solutions of each equation on the interval [0, 2π). 5. cos x = sin x 6. √ 3 cos x tan x - cos x = 0 7. tan2 x + sec x - 1 = 0 8. 1 + cos x = √ 3 sin x Chapter 5 17 Glencoe Precalculus NAME 5-3 DATE PERIOD Practice Solving Trigonometric Equations Solve each equation for all values of x. 1. cos x = 3 cos x - 2 3. √ cos x = 2 cos x - 1 2. 2 sin2 x - 1 = 0 4. 2 sin2 x - 5 sin x + 2 = 0 Find all solutions of each equation on the interval [0, 2π). 5. sec2 x + tan x = 1 6. 3 tan x - √ 3=0 7. 4 sin2 x - 4 sin x + 1 = 0 8. 4 cos2 x - 1 = 0 3 x − 9. cos = cot x sin x 10. tan x sin2 x = 3 tan x Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 11. CIRCLES To find the diameter d of any circle, first inscribe a triangle in the circle. The diameter is then equal to the ratio of any side of the triangle and the sine of its opposite angle. a. Suppose the measure of one side of a triangle inscribed in a circle is 20 centimeters. If the measure of the angle in the triangle opposite this side is 30°, what is the length of the diameter of the circle? b. Suppose a circle with a diameter of 12.4 inches circumscribes a triangle with one side of the triangle measuring 4.6 inches. What is the measure of the angle in the triangle opposite this side? 12. AVIATION An airplane takes off from the ground and reaches a height of 500 feet after flying 2 miles. Given the formula H = d tan θ, where H is the height of the plane and d is the distance (along the ground) the plane has flown, find the angle of ascent θ at which the plane took off. Chapter 5 18 Glencoe Precalculus