10.3B Double Angle Formulas

Double Angle Formulas

Important note: There is an error in this video.

This error occurs when I am finding cos (45º + 45º). (Page 2)

The process is correct but when I checked to see if

the result of the double angle is the same as cos (90º),

I inadvertently wrote sin (90º) = 0 and it should be cos (90º).

Sin Double Angle Formula

Double angles - the two angles have the same value sin (2θ) = 2 sin θ cos θ

Generating the Sum Formula for Sine

~ sin(a + a) = sin a cos a + cos a sin a, resulting in the double angle for sine.

~ sin (2a) = 2 sin a cos a

Example 1

You are given an angle and its location and are asked to find sin(2θ).

1. Know the location of the angle sin θ is in QIV

2. Look at the formula to determine what you need to complete the problem. Find cos θ.

3. Part I Find cos θ using the Pythagorean ID.

4. Determine the sign of cos θ by locating its quadrant.

5. Part II Find sin 2θ

6. Plugin values for sin θ and cos θ

7. Do the math!

Cos(x) Double Angle Formulas

Cosine has three double angle formulas. Each has been derived (at right --->

Cosine Double Angle Hints

When cos x is given, you don't need to solve for cos x because you can use the second cosine double-angle formula. cos(2x) = 2 cos2 x - 1

When sin x is given, you don't need to solve for sin x because you can use the third cosine double angle formula. cos(2x) = 1 - 2sin2x

Example 2

Given the angle and its location.

Find cos 2θ.

1. There are 3 cosine double-angle formulas. We want to do the least amount of work. The formula that has cos x will work for us.

2. Plug in the value for cosine.

3. Do the math!

Example 3

Given the angle, sin θ, and its location, QII.

Find cos 2θ.

1. Again we want to do less work, so we will use the formula with sin θ.

2. Plug in the value for cosine.

3. Do the math!

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