Definition: Double-angle formulas express the sine and cosine of twice an angle in terms of the angle itself, such as sin 2x equals 2 sin x cos x.
What are the formulas?
- sin 2x = 2 sin x cos x
- cos 2x = cos2x – sin2x = 1 – 2 sin2x = 2 cos2x – 1
The three versions of the cosine formula come from the Pythagorean identity. Choose the one that matches the value you know.
How do you use them?
Worked example: suppose sin x = 3/5 with x in quadrant I. Then cos x = 4/5. So sin 2x = 2(3/5)(4/5) = 24/25, and cos 2x = 1 – 2(9/25) = 1 – 18/25 = 7/25.
Check: (24/25)2 + (7/25)2 = 576/625 + 49/625 = 1.
What mistake should you avoid?
Treating 2x as if sin 2x were 2 sin x. Try x = 30 degrees: sin 60 is about 0.866, but 2 sin 30 = 1. Mathbot uses that quick test whenever a formula looks too simple.
Example: sin x = 3/5 gives sin 2x = 24/25 and cos 2x = 7/25