Double Angle Identities: Step-by-Step Guide
Double angle identities let you find the trig functions of 2x using the trig functions of x. sin(2x) = 2 sin(x) cos(x). cos(2x) = cos^2(x) - sin^2(x), which is also 1 - 2 sin^2(x) and 2 cos^2(x) - 1. tan(2x) = 2 tan(x) / (1 - tan^2(x)).
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Pick the identity. Choose sin(2x), cos(2x), or tan(2x). For cosine, pick the form (there are 3) that uses what you already know.\[\cos(2x) = \cos^2 x - \sin^2 x = 1 - 2\sin^2 x = 2\cos^2 x - 1\]
- Find the values you need. Use Pythagorean identities or the given information to find sin(x), cos(x), or tan(x).\[\sin^2 x + \cos^2 x = 1\]
- Use the quadrant for the signs. Use the quadrant of x to decide if sin(x) and cos(x) are positive or negative.\[Q\Rightarrow\text{signs of}\sin,\cos\]
- Put the values in. Substitute into the identity for sin(2x), cos(2x), or tan(2x).\[\sin(2x) = 2\sin x \cos x\]
- Simplify and check. Simplify any roots and check with another form of the identity if you can.\[\sin^2(2x) + \cos^2(2x) = 1\]
Worked example
If sin(x) = 3/5 with x in Quadrant II, find sin(2x) and cos(2x).
Step 1: Use sin(2x) = 2 sin(x) cos(x) and cos(2x) = 1 - 2 sin^2(x).
Step 2: cos^2(x) = 1 - 9/25 = 16/25, so the size of cos(x) is 4/5.
Step 3: In Quadrant II, cosine is negative, so cos(x) = -4/5.
Step 4: sin(2x) = 2 x (3/5) x (-4/5) = -24/25. cos(2x) = 1 - 2 x (9/25) = 1 - 18/25 = 7/25.
Step 5: Check: sin^2(2x) + cos^2(2x) = 576/625 + 49/625 = 625/625 = 1. It works.
The answer is sin(2x) = -24/25 and cos(2x) = 7/25.
Tips and common mistakes
- Learn all three forms of cos(2x), and pick the one that fits what you know.
- sin(2x) = 2 sin(x) cos(x) is used a lot in calculus.
- The quadrant of x tells you the signs. The quadrant of 2x comes out of the answer.
- tan(2x) only works if tan(x) is not 1 or -1 and cos(2x) is not 0.
- These identities help you integrate sin^2 or cos^2 later on.
- You can use them backward: 2 sin(x) cos(x) can be rewritten as sin(2x).
- Power-reduction formulas come from rearranging cos(2x): sin^2(x) = (1 - cos(2x)) / 2, and cos^2(x) = (1 + cos(2x)) / 2.
- Check using sin^2 + cos^2 = 1 on the doubled angle.