Chapter 6 Analytic Trigonometry 6 3 DoubleAngle Power

Chapter 6 Analytic Trigonometry 6. 3 Double-Angle, Power. Reducing, and Half-Angle Formulas Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

Objectives: • Use the double-angle formulas. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2

Double-Angle Formulas Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3

Example: Using Double-Angle Formulas to Find Exact Values If and lies in quadrant II, find the exact value of Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4

Example: Using Double-Angle Formulas to Find Exact Values (continued) If and lies in quadrant II, find the exact value of Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5

Example: Using Double-Angle Formulas to Find Exact Values (continued) If and lies in quadrant II, find the exact value of Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6

Example: Using Double-Angle Formulas to Find Exact Values (continued) If and lies in quadrant II, find the exact value of Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7

Three Forms of the Double-Angle Formula for Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8

Example: Verifying an Identity Verify the identity: Multiply. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9

Example: Verifying an Identity (continued) Verify the identity: continued from previous page Multiply. Simplify. The identity is verified. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11

Chapter 6 Analytic Trigonometry 6. 5 Trigonometric Equations Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12

Objectives: • • Find all solutions of a trigonometric equation. Solve equations with multiple angles. Solve trigonometric equations quadratic in form. Use factoring to separate different functions in trigonometric equations. • Use identities to solve trigonometric equations. • Use a calculator to solve trigonometric equations. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13

Trigonometric Equations and Their Solutions A trigonometric equation is an equation that contains a trigonometric expression with a variable, such as sin x. The values that satisfy such an equation are its solutions. (There are trigonometric equations that have no solution. ) When an equation includes multiple angles, the period of the function plays an important role in ensuring that we do not leave out any solutions. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14

Example: Finding all Solutions of a Trigonometric Equation Solve the equation: Step 1 Isolate the function on one side of the equation. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15

Example: Finding all Solutions of a Trigonometric Equation (continued) Solve the equation: Step 2 Solve for the variable. Solutions for this equation in are: The solutions for this equation are: Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16

Example: Solving an Equation with a Multiple Angle Solve the equation: Because the period is all solutions for this equation are given by Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17

Example: Solving an Equation with a Multiple Angle (continued) Solve the equation: Because the period is all solutions for this equation are given by In the interval , the solutions are: Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18

Example: Solving a Trigonometric Equation Quadratic in Form Solve the equation: The solutions in the interval for this equation are: Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19

Example: Using Factoring to Separate Different Functions Solve the equation: The solutions for this equation in the interval Copyright © 2014, 2010, 2007 Pearson Education, Inc. are: 20

Example: Using an Identity to Solve a Trigonometric Equation Solve the equation: The solutions in the interval Copyright © 2014, 2010, 2007 Pearson Education, Inc. are 21

Example: Solving Trigonometric Equations with a Calculator Solve the equation, correct to four decimal places, for tanx is positive in quadrants I and III In quadrant III The solutions for this equation are 1. 2592 and 4. 4008. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22

Example: Using a Calculator to Solve Trigonometric Equations Solve the equation, correct to four decimal places, for Sin x is negative in quadrants III and IV In quadrant III In quadrant IV The solutions for this equation are 3. 3752 and 6. 0496. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23