Page 1 of 2 14.7 Using Double- and Half-Angle …

1 Page 1 of 2. Using double - and Half-Angle Formulas GOAL 1 double - AND Half-Angle FORMULAS. What you should learn GOAL 1 Evaluate In this lesson you will use formulas for double angles (angles of measure 2u). expressions Using double - and Half-Angle formulas. u . and half angles angles of measure . The three formulas for cos 2u below are 2. u equivalent, as are the two formulas for tan . Use whichever formula is most GOAL 2 Use double - and 2. Half-Angle formulas to solve convenient for solving a problem. real-life problems, such as finding the mach number for Double- ANGLE AND Half-Angle FORMULAS. an airplane in Ex.

2 70. Why you should learn it Double- ANGLE FORMULAS. cos 2u = cos2 u sin2 u sin 2u = 2 sin u cos u To model real-life situations with double - and cos 2u = 2 cos2 u 1 2 tan u tan 2u = . 2. Half-Angle relationships, such 1 tan u as kicking a football cos 2u = 1 2 sin2 u in Example 8. AL LI Half-Angle FORMULAS. FE. RE. u 1 c2o s u sin = . 2. u tan = . 2. 1 cos u sin u u 1 + c2o s u cos = . 2. u tan = . 2. sin u 1 + cos u u u u The signs of sin and cos depend on the quadrant in which lies. 2 2 2. EXAMPLE 1 Evaluating Trigonometric Expressions . Find the exact value of (a) tan and (b) cos 105 . 8. SOLUTION.. a. Use the fact that is half of.

3 8 4.. 2. 1 1 cos . 4 = 2 = = 2. 1 2 2 . STUDENT HELP tan = tan = . 1. 8 2 4 sin . 2. 2 . Study Tip 4 2. In Example 1 note that, in u 1 b. Use the fact that 105 is half of 210 and that cosine is negative in Quadrant II. general, tan tan u. 2 2. 1 + co 2s 2 1 0 . Similar statements can 1. be made for the other cos 105 = cos (210 ) = . 2. trigonometric functions 1 + . 2 . = = . of double and half 3 .. angles. 2 3 . = . 2 .. 3 . 2 4 2. Using double - and Half-Angle Formulas 875. Page 1 of 2. STUDENT HELP EXAMPLE 2 Evaluating Trigonometric Expressions Study Tip 3 3 . Because < u < in 3 Given cos u = with < u < , find the following.

4 2 5 2. Example 2, you can u multiply through the a. sin 2u b. sin . 2. 1. inequality by to get 2 SOLUTION. u 3 u < < , so is in 4. 2 2 4 2 a. Use a Pythagorean identity to conclude that sin u = . 5. Quadrant II. sin 2u = 2 sin u cos u 45 35 . = 2 = . 24. 25. u u b. Because is in Quadrant II, sin is positive. 2 2. 1 .. 3.. = . u 1 co s u 5 4 2 5. sin = = = . 2 2 2 5 5. EXAMPLE 3 Simplifying a Trigonometric Expression cos 2 . Simplify . sin + cos . SOLUTION. STUDENT HELP. co s 2 cos2 sin2 . = Use a Double- angle formula . Study Tip sin + cos sin + cos . Because there are three (cos sin )(cos + sin ). formulas for cos 2u, you = Factor difference of squares.

5 Will want to choose the sin + cos . one that allows you to = cos sin Simplify. simplify the expression in which cos 2u appears, as illustrated in Example 3. EXAMPLE 4 Verifying a Trigonometric Identity Verify the identity sin 3x = 3 sin x 4 sin3 x. SOLUTION. sin 3x = sin (2x + x) Rewrite sin 3x as sin (2x + x). = sin 2x cos x + cos 2x sin x Use a sum formula . = (2 sin x cos x) cos x + (1 2 sin2 x) sin x Use Double- angle formulas. = 2 sin x cos2 x + sin x 2 sin3 x Multiply. = 2 sin x (1 sin2 x) + sin x 2 sin3 x Use a Pythagorean identity. = 2 sin x 2 sin3 x + sin x 2 sin3 x Distributive property = 3 sin x 4 sin3 x Combine like terms.

6 876 Chapter 14 Trigonometric Graphs, Identities, and Equations Page 1 of 2. EXAMPLE 5 Solving a Trigonometric Equation STUDENT HELP Solve tan 2x + tan x = 0 for 0 x < 2 . NE. ER T. HOMEWORK HELP. INT. Visit our Web site SOLUTION. tan 2x + tan x = 0 Write original equation. for extra examples. 2 tan x + tan x = 0 Use a Double- angle formula . 1 tan2 x 2 tan x + tan x (1 tan2 x) = 0 Multiply each side by 1 tan2 x. 2 tan x + tan x tan3 x = 0 Distributive property. 3. 3 tan x tan x = 0 Combine like terms. tan x (3 tan2 x) = 0 Factor. Set each factor equal to 0 and solve for x. tan x = 0 or 3 tan2 x = 0. x = 0, 3 = tan2 x 3 = tan x 2 4 5.

7 X = , , , . 3 3 3 3. CHECK You can use a graphing calculator to check the solutions. Graph the following function: y = tan 2x + tan x Then use the Zero feature to find the x-values for Zero which y = 0. X= Y=0.. Some equations that involve double or half angles can be solved directly without resorting to double - or Half-Angle formulas. EXAMPLE 6 Solving a Trigonometric Equation x Solve 2 cos + 1 = 0. 2. SOLUTION. x 2 cos + 1 = 0 Write original equation. 2. x 2 cos = 1 Subtract 1 from each side. 2. x 1. cos = Divide each side by 2. 2 2. x 2 4 x = + 2n or + 2n General solution for }}. 2. 2 3 3. 4 8 . x = + 4n or + 4n General solution for x 3 3.

8 Using double - and Half-Angle Formulas 877. Page 1 of 2. GOAL 2 Using TRIGONOMETRY IN REAL LIFE. The path traveled by an object that is projected at an initial height of h0 feet, an initial speed of v feet per second, and an initial angle is given by 16. y = . 2.. 2. x2 + (tan )x + h0. v cos . where x and y are measured in feet. (This model neglects air resistance.). EXAMPLE 7 Simplifying a Trigonometric Model SPORTS Find the horizontal distance traveled by a football kicked from ground level (h 0 = 0) at speed v and angle .. SOLUTION Not drawn to scale Using the model above with h 0 = 0, set y equal to 0 and solve for x.

9 16.. 2.. 2. x2 + (tan )x = 0 Let y = 0. v co s . ( x) 16. x tan = 0. v2 c o s 2 . Factor. 16. x tan = 0 Zero product property v2 cos2 (Ignore x = 0.). 16. x = tan Add tan to each side. v2 c o s 2 . 1 1. x = v2 cos2 tan Multiply each side by }}v 2 cos2 . 16. 16. 1. x = v2 cos sin Use cos tan = sin . 16. FOCUS ON 1 1 1. APPLICATIONS x = v2 (2 cos sin ) Rewrite }} as }} 2. 16 32. 32. 1. x = v2 sin 2 Use a Double- angle formula . 32. EXAMPLE 8 Using a Trigonometric Model SPORTS You are kicking a football from ground level with an initial speed of 80 feet per second. Can you make the ball travel 200 feet? L. AL I.

10 SOLUTION. FIELD GOALS The FE. RE. 1. longest professional 200 = (80)2 sin 2 Substitute for x and v in the formula from Example 7. field goal was 63 yards, 32. made by Tom Dempsy in 1. 1 = sin 2 Divide each side by }}(80)2 = 200. 1970. This record was tied 32. by Jason Elam during the 90 = 2 sin 1 1 = 90 . 1998 1999 season. NE. ER T 45 = Solve for . INT. APPLICATION LINK. You can make the football travel 200 feet if you kick it at an angle of 45 . 878 Chapter 14 Trigonometric Graphs, Identities, and Equations Page 1 of 2. GUIDED PRACTICE. Vocabulary Check 1. Complete this statement: sin 2u = 2 sin u cos u is called the.