10-3

10-3 Double-Angle and Half-Angle Formulas

We can use ten different trigonomic formulas in this section:

1. sin 2(a) = 2 sin(a) cos(a)



2. cos 2(a) = cos^2(a) - sin^2(a)



3. cos 2(a) = 1 - 2 sin^2(a)



4. cos 2(a) = 2 cos^2(a) - 1



5. tan 2(a) = 2 tan(a)/1 - tan^2(a)



6. sin a/2 = +/- square root of (1 - cos(a)/2)



7. cos a/2 = +/- square root of (1 + cos(a)/2)



8. tan a/2 = +/- square root of (1 - cos(a)/1 + cos(a))



9. tan a/2 = sin(a)/1 + cos(a)



10. tan a/2 = 1 - cos(a)/sin(a)

Example 1: Simplify the given expression.

1) 2 cos^2 10 - 1

• cos 2(10)



• cos(20)

2) cos^2 4A - sin^2 4A

• cos 2(4A)



• cos (8A)

Example 2: Find the exact value of the given expression.

1) 1 - 2 sin^2(7pi/12)

• cos 2(105)



• cos(210)



• Now you must find a reference angle: 210 - 180 = 30



• square root of 3/2

Example 3: If sin A = 5/13, find sin 2A.

• sin 2A = 2 sin A cos A



• sin 2A = 2 (5/13) (12/13)



• (10/13) (12/13)



• 120/169

-Jordan Duhon