The Double Angle Formulas

Also known as double angle identities, there are three distinct double angle formulas: sine, cosine, and tangent. The cosine double angle formula has three variations. The double angle formulas relate different trigonometry functions to each other.

The three double angle formulas and their variations are given as:

• sin(2θ) = 2sin(θ)cos(θ)
• cos(2θ) = cos²(θ) - sin²(θ)
• cos(2θ) = 2cos²(θ) - 1
• cos(2θ) = 1 - 2sin²(θ)
• tan(2θ) = ^[2tan(θ)]⁄_{[1 - tan²(θ)]}

When to use the Formulas

Each double angle formula is useful for simplifying expressions that contain trigonometric terms. For example, an expression may have the term "cos(2θ)" in it. We can substitute the term "1 - 2sin²(θ)" in for "cos(2θ)". This will help us if we need the expression to be in terms of sine.

Here's another instance of using a double angle formula to simplify an expression. Imagine coming across the expression "sin²(θ) - cos²(θ)". Not too bad, but having to hand calculate the value of this expression by plugging in angles could be difficult. The first cosine formula tells us that we can replace "sin²(θ) - cos²(θ)" with "-cos(2θ)". It is now is much simpler and more manageable to hand calculate.

Double Angle Formula Example Problem

Prove the sine double angle formula by using the angle θ = 4 radians.

Solution:

1. sin(4) = -0.757
2. Let's plug in our values to the double angle formula.
   sin(2·4) = 2sin(4)cos(4)
   0.989 = 2(-0.757)(-0.654)
3. 0.989 = 0.989 √