Double-Angle and Half-Angle Formulas

1. Double-Angle and Half-Angle Formulas Dr .Hayk Melikyan Departmen of Mathematics and CS melikyan@nccu.edu

2. Double-Angle Identities

3. Three Forms of the Double-Angle Formula for cos2

4. Power-Reducing Formulas

5. Example Write an equivalent expression for sin4x that does not contain powers of trigonometric functions greater than 1. Solution

6. Half-Angle Identities

7. Text Example Find the exact value of cos 112.5°. Solution Because 112.5° =225°/2, we use the half-angle formula for cos /2 with = 225°. What sign should we use when we apply the formula? Because 112.5° lies in quadrant II, where only the sine and cosecant are positive, cos 112.5° < 0. Thus, we use the - sign in the half-angle formula.

8. Half-Angle Formulas for:

9. Verifying a Trigonometric Identity Verify the identity:

10. The Half-Angle Formulas for Tangent

11. Example • Verify the following identity: Solution

12. Product-to-Sum and Sum-to-Product Formulas Product-to-Sum Formulas

13. Example Solution • Express the following product as a sum or difference:

14. sin  sin  = 1/2 [cos( - ) - cos( + )] sin  cos  = 1/2[sin( + ) + sin( - )] Express each of the following products as a sum or difference. a. sin 8x sin 3xb. sin 4x cos x Solution The product-to-sum formula that we are using is shown in each of the voice balloons. a. sin 8x sin 3x= 1/2[cos (8x- 3x) - cos(8x+ 3x)] = 1/2(cos 5x- cos 11x) Text Example b. sin 4x cos x= 1/2[sin (4x+x) + sin(4x-x)] = 1/2(sin 5x+ sin 3x)

15. Evaluating the Product of a Trigonometric Expression Determine the exact value of the expression

16. Sum-to-Product Formulas

17. Example • Express the difference as a product: Solution

18. Example • Express the sum as a product: Solution

19. Example • Verify the following identity: Solution

20. Example

21. Example

22. Equations Involving a Single Trigonometric Function • To solve an equation containing a single trigonometric function: • • Isolate the function on one side of the equation. • sinx = a (-1 ≤ a ≤ 1 ) • cosx = a (-1 ≤ a ≤ 1 ) • tan x = a ( for any real a ) • • Solve for the variable.

23. Trigonometric Equations x y y = cos x 1 y = 0.5 x –4  –2  2  4  –1 cos x = 0.5 has infinitely many solutions for –< x <  y y = cos x 1 0.5 2  cos x = 0.5 has two solutions for 0 < x < 2 –1

24. This is the given equation. 3 sin x- 2 = 5 sin x- 1 Subtract 5 sin x from both sides. 3 sin x- 5 sin x- 2 = 5 sin x- 5 sin x – 1 Simplify. -2 sin x- 2 =-1 Add 2 to both sides. -2 sin x= 1 Divide both sides by -2 and solve for sin x. sin x= -1/2 Text Example Solve the equation: 3 sin x- 2 = 5 sin x- 1. Solution The equation contains a single trigonometric function, sin x. Step 1Isolate the function on one side of the equation. We can solve for sin x by collecting all terms with sin x on the left side, and all the constant terms on the right side.

25. Solution The given equation is in quadratic form 2t2+t- 1 = 0 with t= cos x. Let us attempt to solve the equation using factoring. This is the given equation. 2 cos2x+ cos x- 1 = 0 Factor. Notice that 2t2 + t – 1 factors as (t – 1)(2t + 1). (2 cos x- 1)(cos x+ 1) = 0 Set each factor equal to 0. 2 cos x- 1= 0 or cos x+ 1 = 0 2 cos x= 1 cos x= -1 Solve for cos x. Text Example Solve the equation: 2 cos2 x+ cos x- 1 = 0, 0 £x< 2p. cos x= 1/2 x=px= 2pppx=p The solutions in the interval [0, 2p) are p/3, p, and 5p/3.

26. Example • Solve the following equation: Solution:

27. Example • Solve the equation on the interval [0,2) Solution:

28. Example Solve the equation on the interval [0,2) Solution:

29. Example • Solve the equation on the interval [0,2) Solution: