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

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

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

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 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. 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 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. 8 Three Forms of the Double-Angle Formula for

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

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

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

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

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 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. 14 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. 15 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. 16 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. 17 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. 18 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. 19 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. 20 Example: Using Factoring to Separate Different Functions Solve the equation: The solutions for this equation in the interval are:

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

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 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 I In quadrant III The solutions for this equation are and

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23 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 and