What Are Double Angle Identities?

Once you start doing trigonometry and calculus, you may run into trigonometric functions like sine, cosine, and tangent. Playing trial and error with charts or a calculator to find the answer to trigonometric equations would range from a drawn-out nightmare to totally impossible. The many trig identities and relationships become crucial when solving for these trigonometric ratios. The double-angle identities are special instances of what's known as a compound formula, which breaks functions of the forms (A + B) or (A – B) down into functions of either A or B.

There are many other trigonometric identities that you might recognize. The pythagorean identity for a right triangle, half-angle formulas, sum formula, and difference identities are all very useful relationships.

There are three double-angle identities, one each for the sine, cosine and tangent functions. The sine and cosine functions can both be written with multiple special cases.

The Double-Angle Identities for Sine

Here are the two ways of writing the double-angle identity for the sine function:

\(\begin{align*}\)
\(\sin(2\theta)\)
\(&= 2\sin(\theta)\cos(\theta)\)
\(&= \frac{2\tan{\theta}}{1 + \tan^2\theta}\)
\(\end{align*}\)

The Double-Angle Identities for Cosine

There are even more ways of writing the double-angle identity for cosine:

\(\begin{align*}\)
\(\cos{2\theta}\)
\(&= \cos^2(\theta) – \sin^2(\theta)\)
\(&= 2\cos^2(\theta) – 1\)
\(&= 1 – 2\sin^2(\theta)\)
\(&= \frac{1 – \tan^2(\theta)}{1 + \tan^2(\theta)}\)
\(\end{align*}\)

The Double-Angle Identity for Tangent

There is just one practical way to write the double-angle identity for the tangent function:

\(\begin{align*}\)
\(\tan(2\theta)\)
\(&= \frac{2\tan(\theta)}{1-\tan^2(\theta)}\)
\(\end{align*}\)

Using Double-Angle Identities

There are numerous trigonometric expressions and scenarios to solve. These formulas will often use theta (Θ), x or alpha (ɑ), but regardless of the variable, the double angle formula will be able to find an equivalent exact value for a suitable situation.

Example 1

Suppose we want to find an equivalent statement in terms of sin θ and cos θ to the following trig function:

\(\cos2x + \sin2x\)

We can use the double angle formulas to reduce this expression to only use sin θ and cos θ:

\(\begin{align*}\)
\((2\cos^2(x) – 1) + (2\sin(x)\cos(x))\)
\(&= (2\cos^2(x) + 2\sin(x)\cos(x)) – 1\)
\(& = 2\cos(x)(\cos(x) + \sin(x)) – 1\)
\(\end{align*}\)

Example 2

Simplify the following expressions:

\(2\cos^2(32) -1\)
\(\ \text{}\)
\(2\sin α \cos α \ \text{ where } α = \frac{1}{2}\beta\)

Using the cosine double-angle identity on the first expression we can just use the value of cos to represent the same value:

\(\begin{align*}\)
\(2\cos^2(32)-1\)
\(&=\cos{2 \times 32}\)
\(&=\cos{64}\)
\(\end{align*}\)

We can apply a similar process to the second expression using the sine double-angle identity:

\(\begin{align*}\)
\(2\sin{α}\cos{α}\)
\(&= \sin{2 \times α}\)
\(&= \sin{2 \times \frac{\beta}{2}}\)
\(&= \sin{\beta}\)
\(\end{align*}\)