Once you start doing trigonometry and calculus, you may run into expressions like sin(2θ), where you're asked to find the value of θ. Playing trial and error with charts or a calculator to find the answer would range from a drawn-out nightmare to totally impossible. Fortunately, the double-angle identities are here to help. These 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 just A and B.
The Double-Angle Identities for Sine
There are three double-angle identities, one each for the sine, cosine and tangent functions. But the sine and cosine identities can be written in multiple ways. Here are the two ways of writing the double-angle identity for the sine function:
- sin(2θ) = 2sinθcosθ
- sin(2θ) = (2tanθ)/(1 + tan2θ)
The Double-Angle Identities for Cosine
There are even more ways of writing the double-angle identity for cosine:
- cos(2θ) = cos2θ – sin2θ
- cos(2θ) = 2cos2θ – 1
- cos(2θ) = 1 – 2sin2θ
- cos(2θ) = (1 – tan2θ)/(1 + tan2θ)
The Double-Angle Identity for Tangent
Mercifully, there is just one way to write the double-angle identity for the tangent function:
- tan(2θ) = (2tanθ)/(1 – tan2θ)
Using Double-Angle Identities
Imagine that you're faced with a right triangle where you know the length of its sides, but not the measure of its angles. You've been asked to find θ, where θ is one of the triangle's angles. If the hypotenuse of the triangle measures 10 units, the side adjacent to your angle measures 6 units and the side opposite the angle measures 8 units, it doesn't matter that you don't know the measure of θ; you can use your knowledge of sine and cosine, plus one of the double-angle formulas, to find the answer.
Find Sine and Cosine
Choose a Double-Angle Formula
Substitute in Known Values
Convert to Decimal Form
Find the Inverse Sine
Solve for θ
Once you've chosen an angle, you can define sine as the ratio of the opposite side over the hypotenuse, and cosine as the ratio of the adjacent side over the hypotenuse. So in the example just given, you have:
sinθ = 8/10
cosθ = 6/10
You find these two expressions because they're the most important building blocks for the double-angle formulas.
Because there are so many double-angle formulas to choose from, you can select the one that looks easier to compute and will return the type of information you need. In this case, because you know sinθ and cosθ already, sin(2θ) = 2sinθcosθ looks convenient.
You already know the values of sinθ and cosθ, so substitute them into the equation:
sin(2θ) = 2(8/10)(6/10)
Once you simplify, you'll have:
sin(2θ) = 96/100
Most trigonometric charts are given in decimals, so next work the division represented by the fraction to convert it to decimal form. Now you have:
sin(2θ) = 0.96
Finally, find the inverse sine or arcsine of 0.96, which is written as sin-1(0.96). Or, in other words, use your calculator or a chart to approximate the angle that has a sine of 0.96. As it turns out, that is almost exactly equal to 73.7 degrees. So 2θ = 73.7 degrees.
Divide each side of the equation by 2. This gives you:
θ = 36.85 degrees