Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Trigonometric Equations Learn to solve trigonometric equations of the form a sin ( x – c ) = k, a cos ( x – c ) = k, and a tan ( x – c ) = k. Learn to solve trigonometric equations involving multiple angles. Learn to solve trigonometric equations by using the zero-product property. Learn to solve trigonometric equations that involve more than one trigonometric function. Learn to solve trigonometric equations by squaring both sides. SECTION

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley TRIGONOMETRIC EQUATIONS A trigonometric equation is an equation that contains a trigonometric function with a variable. Equations that are true for all values in the domain of the variable are called identities. Equations that are true for some but not all values of the variable are called conditional equations. Solving a trigonometric equation means to find its solution set.

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solving a Linear Trigonometric Equation Find all solutions in the interval [0, 2π) of the equation: Solution Replace by  in the given equation. We know sin  is (+) in Q I and II

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solution continued Solving a Linear Trigonometric Equation andor Solution set in [0, 2π] is

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solving a Trigonometric Equation Containing Multiple Angles Find all solutions in the interval [0, 2π) of the equation: Period of cosine function is 2π. Replace  with 3x. cos  is (+) in Q I and IV, Solution Recall Soor so

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solution continued Solving a Linear Trigonometric Equation Oror To find solutions in the interval [0, 2π), try: n = –1 n = 0 n = 1

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solution continued Solving a Linear Trigonometric Equation Solution set is Values resulting from n = –1 are too small. n = 2 n = 3 Values resulting from n = 3 are too large. Solutions we want correspond to n = 0, 1, and 2.

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Solving a Quadratic Trigonometric Equation Find all solutions of the equation Express the solutions in radians. Solution Factor No solution because –1 ≤ sin  ≤ 1.

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Solution continued So, Since sin  has period 2π, the solutions of the given equation are where n is any integer. are the only two solutions in the interval [0, 2π). Solving a Quadratic Trigonometric Equation

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Solving a Trigonometric Equation Using Identities Find all the solutions in the interval [0, 2π) to the equation Solution Use the Pythagorean identity to rewrite the equation in terms of cosine only.

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Solution continued Solving a Trigonometric Equation Using Identities Use the quadratic formula to solve this equation.

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Solution continued Solving a Trigonometric Equation Using Identities Hence, No solution because –1 ≤ cos  ≤ 1. cos  is (–) in QII, QIII

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Solution continued Solving a Trigonometric Equation Using Identities Solution set in the interval [0, 2π) is

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Solving a Trigonometric Equation by Squaring Solution Square both sides and use identities to convert to an equation containing only sin x. Find all the solutions in the interval [0, 2π) to the equation

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Solution continued Solving a Trigonometric Equation by Squaring

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Solution continued Possible solutions are: Solving a Trigonometric Equation by Squaring Solution set in the interval [0, 2π) is  