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.
TL;DR (Too Long; Didn't Read)
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:
The Double-Angle Identities for Cosine
There are even more ways of writing the double-angle identity for cosine:
The Double-Angle Identity for Tangent
There is just one practical way to write the double-angle identity for the tangent function:
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.
Tips
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:
We can use the double angle formulas to reduce this expression to only use sin θ and cos θ:
Example 2
Simplify the following expressions:
Using the cosine double-angle identity on the first expression we can just use the value of cos to represent the same value:
We can apply a similar process to the second expression using the sine double-angle identity:
References
About the Author
Lisa studied mathematics at the University of Alaska, Anchorage, and spent several years tutoring high school and university students through scary -- but fun! -- math subjects like algebra and calculus.