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Chapter 5 Analytic Trigonometry © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved 1 SECTION 5.4 Double-Angle and Half-Angle Formulas OBJECTIVES 1 2 3 4 Use double-angle formulas. Use power-reducing formulas. Use half-angle formulas. Solve trigonometric equations involving multiple angles and half-angles. © 2010 Pearson Education, Inc. All rights reserved 2 DOUBLE-ANGLE FORMULAS sin 2x 2sin x cos x cos 2x cos x sin x 2 2 cos 2x 1 2sin x 2 2 tan x tan 2x 2 1 tan x cos 2x 2 cos x 1 © 2010 Pearson Education, Inc. All rights reserved 2 3 EXAMPLE 1 Using Double-Angle Formulas 3 If cos and is in quadrant II, find the 5 exact value of each expression. a. sin 2 b. cos 2 c. tan 2 Solution Use identities to find sin θ and tan θ. 9 4 θ is in QII so 2 sin 1 cos 1 25 5 sin > 0. sin 4/5 4 tan cos 3 / 5 3 © 2010 Pearson Education, Inc. All rights reserved 4 EXAMPLE 1 Using Double-Angle Formulas Solution continued a. sin 2 2sin cos b. cos 2 cos sin 4 3 2 5 5 24 25 © 2010 Pearson Education, Inc. All rights reserved 2 2 2 3 4 5 5 9 16 25 25 7 25 2 5 EXAMPLE 1 Using Double-Angle Formulas Solution continued 2 tan c. tan 2 1 tan 2 8 3 16 1 9 8 3 7 9 4 2 3 2 4 1 3 8 9 3 7 © 2010 Pearson Education, Inc. All rights reserved 24 7 6 © 2010 Pearson Education, Inc. All rights reserved 7 © 2010 Pearson Education, Inc. All rights reserved 8 © 2010 Pearson Education, Inc. All rights reserved 9 © 2010 Pearson Education, Inc. All rights reserved 10 © 2010 Pearson Education, Inc. All rights reserved 11 © 2010 Pearson Education, Inc. All rights reserved 12 © 2010 Pearson Education, Inc. All rights reserved 13 EXAMPLE 3 Finding a Triple-Angle Formula for Sine Verify the identity sin 3x = 3 sin x – 4 sin3 x. We choose the sine-squared form since we’re looking for a sine-cubed term. Solution sin 3x = sin (2x + x) = sin 2x cos x + cos 2x sin x = (2 sin x cos x) cos x + (1 – 2 sin2 x) sin x = 2 sin x cos2 x + sin x – 2 sin3 x = 2 sin x (1 – sin2 x) + sin x – 2 sin3 x = 2 sin x – 2 sin3 x + sin x – 2 sin3 x = 3 sin x – 4 sin3 x © 2010 Pearson Education, Inc. All rights reserved 14 © 2010 Pearson Education, Inc. All rights reserved 15 Omit: POWER REDUCING FORMULAS 1 cos 2x sin x 2 2 1 cos 2x cos x 2 2 1 cos 2x tan x 1 cos 2x 2 © 2010 Pearson Education, Inc. All rights reserved 16 Examples 4 and 5 omitted © 2010 Pearson Education, Inc. All rights reserved 17 HALF-ANGLE FORMULAS 1 cos sin 2 2 1 cos cos 2 2 1 cos tan 2 1 cos The sign, + or –, depends on the quadrant in which lies. (Know this detail about sign.) 2 You will be provided with these three half-angle formulas. © 2010 Pearson Education, Inc. All rights reserved 18 EXAMPLE 6 Using Half-Angle Formulas Use a half-angle formula to find the exact value of cos 157.5º. Solution 315º Because 157.5º , use half-angle formula for 2 cos with 315º . 157.5º is in QII, cos 0. 2 2 Important 315º cos157.5º cos 2 1 cos315º 2 © 2010 Pearson Education, Inc. All rights reserved 19 EXAMPLE 6 Using Half-Angle Formulas Solution continued 1 cos 45º 2 2 2 22 2 1 2 2 22 2 2 2 © 2010 Pearson Education, Inc. All rights reserved 20 © 2010 Pearson Education, Inc. All rights reserved 21 © 2010 Pearson Education, Inc. All rights reserved 22 © 2010 Pearson Education, Inc. All rights reserved 23 Omit Example 8. © 2010 Pearson Education, Inc. All rights reserved 24 EXAMPLE 9 Solving a Trigonometric Equation Involving Multiple Angles Find all solutions in the interval [0, 2π) of the equation 2 sin sin 3 0. Solution 2sin sin 3 0 2sin 3sin 4sin 0 3 2 sin 3sin 4 sin 0 4 sin 3 sin 0 3 sin 4sin 2 1 0 sin 2sin 1 2sin 1 0 © 2010 Pearson Education, Inc. All rights reserved 25 EXAMPLE 9 Solving a Trigonometric Equation Involving Multiple Angles Solution continued sin 0 or 2sin 1 0 or 1 sin 0 sin 2 0 or 6 or 2sin 1 0 1 sin 2 7 = or 6 6 11 2 6 6 5 6 6 5 7 11 Solution set is 0, , , , . 6 6 6 6 © 2010 Pearson Education, Inc. All rights reserved 26 © 2010 Pearson Education, Inc. All rights reserved 27 EXAMPLE 10 Solving a Trigonometric Equation Involving Half-Angles Find all solutions in the interval [0, 2π) of the x equation sin x cos 0. 2 Solution For x to be a solution in the interval [0, 2π), the x value of must be in the interval [0, π). 2 x sin x cos 0. 2 x x x 2sin cos cos 0 2 2 2 x x cos 2 sin 1 0 2 2 © 2010 Pearson Education, Inc. All rights reserved 28 EXAMPLE 10 Solving a Trigonometric Equation Involving Half-Angles Solution continued x cos 0 2 x 2 sin 1 0 or 2 x 1 x sin 2 2 2 2 x x 5 or x 2 6 2 6 5 x or x 3 3 5 Solution set is , , . 3 3 © 2010 Pearson Education, Inc. All rights reserved 29 When using a graphing utility, restrict the domain to the interval of interest, as in the example using this problem . . © 2010 Pearson Education, Inc. All rights reserved 30 We are omitting Section 5.5 Product-to-Sum and Sum-to-Product Formulas. Beside the omitted power reduction formulas and the Sum-and-Difference and Half-Angle Formulas that you will be given, you are to commit to memory the remaining formulas on page 386. You are not responsible for knowing the eight formulas listed at the top of page 387 (from section 5.5). © 2010 Pearson Education, Inc. All rights reserved 31 the text on the left, These are the only formulas you will be given for Test I (and the FE for this portion of the material). They are placed together on the final page. You must know or be able to derive any of the others you may need. Recall that we had several from Chapter 4, also. © 2010 Pearson Education, Inc. All rights reserved 32