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.

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These identities can also be called theorems, which refers to a slightly different item in math, but functionally they describe the same thing. Some of these identities also have equivalent names (half-angle identities, sum identities, addition formulas, etc.).

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

The Double-Angle Identities for Cosine

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

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

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The variable ‌_x_‌ was commonly used in all algebraic problems in early algebra and precalculus, but when we look at sin ‌_x_‌ cos ‌_x_‌ tan x this is equivalent to using theta or alpha, such as sin α cos α tan α. All variables simply keep track of possible values and their position on each side of the equation.

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

Example 2

Simplify the following expressions:

(2cos^2(32) -1)( text{})(2sin 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*})(2cos^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*})(2sin{α}cos{α})(&= sin{2 times α})(&= sin{2 times frac{beta}{2}})(&= sin{beta})(end{align*})