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Section 5.3 Double-Angle, Power-Reducing, and Half-Angle Formulas Objectives: • Use the double-angle formulas. • Use the power-reducing formulas. • Use the half-angle formulas. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 Double-Angle Formulas Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Example1 Part A: Using Double-Angle Formulas to Find Exact Values 4 If sin and 5 lies in quadrant II, find the exact value of sin 2 . Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Example 1A Continued If sin 4 and lies in quadrant II, find the exact value 5 of sin 2 . sin 2 2sin cos x 3 y4 r 5 r 5 4 4 3 2 5 5 24 25 x y 4 sin r 5 x 3 cos r 5 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Example1 Part B 4 If sin and 5 lies in quadrant II, find the exact value of cos 2 . Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example1B Continued If sin 4 and lies in quadrant II, find the exact value 5 of cos 2 . cos 2 cos 2 sin 2 2 x 3 y4 r 5 y 4 sin r 5 x 3 cos r 5 3 4 5 5 9 16 25 25 7 25 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 6 Example1 Part C: Using Double-Angle Formulas to Find Exact Values 4 If sin and 5 lies in quadrant II, find the exact value of tan 2 . Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example 1 C Continued If sin 4 and lies in quadrant II, find the exact value 5 of tan 2 . 2 tan tan 2 2 1 tan x 3 4 y4 2 3 r 5 2 4 1 y 4 3 sin y 4 r 5 tan Continued x 3 x 3 cos r 5 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example 1 C Continued Continued 2 tan tan 2 1 tan 2 4 8 8 2 3 3 3 2 7 4 16 1 1 9 9 3 8 9 3 7 24 7 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example 2 Find the exact value of the trigonometric expression 2sin165 cos165 2sin165 cos165 sin[2(165)] sin 330 1 2 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Three Forms of the Double-Angle Formula for cos 2 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Example3: Verifying an Identity 3 sin 3 3sin 4sin Verify the identity: now expand using sin(A+B) sin(2 ) sin 2 cos cos 2 sin sin 2 2sin cos 2sin cos cos (1 2sin ) sin 2 cos 2 1 2sin 2 2sin cos 2 sin 2sin 3 2sin (1 sin 2 ) sin 2sin 3 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example 3 Continued Cont. 2sin (1 sin 2 ) sin 2sin 3 2sin 2sin 3 sin 2sin 3 3sin 4sin 3 distribute Combine like terms Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Half-Angle Formulas Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example 4: Using the Half-Angle Formula to Find an Exact Value 3 Use cos 210 to find the exact value of cos105°. 2 3 210 1 cos105 cos 2 2 2 1 cos cos 2 2 210 1 cos 210 cos 2 2 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example 4 Continued 3 Use cos 210 to find the exact value of cos105°. 2 3 210 1 cos 210 1 cos 2 2 2 2 2 2 3 1 2 2 3 4 2 2 3 2 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Example 5 Find sin α/2, cos α/2, and tan α/2 if sec α = -5/4 and α lies in Quadrant II. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Half-Angle Formulas for Tangent Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Power-Reducing Formulas - Look at but won’t assign Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Reducing the Power of a Trigonometric Function Write an equivalent expression for sin4 x that does not contain powers of trigonometric functions greater than 1. 2 2 1 cos 2 x 1 2cos 2 x cos 2 x 2 4 2 sin x sin x 4 2 1 1 1 1 cos 2(2 x) 1 1 1 2 cos 2 x cos 2 x cos 2 x 4 2 4 2 4 2 4 1 1 1 1 1 1 1 cos 2 x 1 cos 4 x cos 2 x cos 4 x 4 2 8 4 2 8 8 3 1 1 cos 2 x cos 4 x 8 2 8 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20