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Double Angle Identities Calculator

Created by Davide Borchia
Reviewed by Komal Rafay
Last updated: Jan 18, 2024


Knowing how to calculate the double angle identities will help you with countless problems after your first encounter with trigonometry. From high school to university, and even after, this handy tool will give you the help you need, refreshing or calculating these critical mathematical formulas.
With this tool, you will learn the following:

  • What are double angle trig identities;
  • How to calculate the double angle identities in trigonometry;

and much more, as examples and related tools!

What are trigonometric double angle identities?

Double angle identities are a class of trigonometric identities (that is, an equality that relates two mathematical formulas, being valid for all the values in a given range) which equate the value of a trigonometric function for twice the value of an angle to an algebraic combination of other trigonometric functions applied to the value of the angle.

Double angle identities allow you to calculate the value of functions such as sin(2α)\sin(2\alpha), cos(4β)\cos(4\beta), and so on. This class of identities is a particular case of the compound angle identities, which allow you to calculate the trigonometric functions of sums of angles.

How do I calculate the double angle identities?

In this section, we will learn how to calculate the double angle identities for the three fundamental trigonometric functions (sine, cosine, and tangent). Let's see them one by one!

Calculate the double angle identity for the sine

The double angle identity for the sine is the first one we will meet in our journey. In mathematical terms, we can use the following formula:

sin(2α)=2sin(α)cos(α)\scriptsize \sin(2\alpha) = 2\sin(\alpha)\cos(\alpha)

Using other trigonometric identities, we can write the formula above in different ways.

  • Using the relationship between sine and cosine (the Pythagorean identity), sin2(α)+cos2(α)=1\sin^2(\alpha)+\cos^2(\alpha) = 1, and the formula for the square of a polynomial ((a+b)2=a2+b2+2ab(a+b)^2 = a^2+b^2+2ab), we can find another way to write the right-hand side of the identity:
(sin(α)+cos(α))2=sin2(α)+cos2(α)+2sin(α)cos(α)2sin(α)cos(α)=(sin(α)+cos(α))2sin2(α)cos2(α)2sin(α)cos(α)=(sin(α)+cos(α))21\scriptsize\quad\begin{split} \left(\sin(\alpha)+\cos(\alpha)\right)^2&=\sin^2(\alpha)+\cos^2(\alpha)\\ &+2\sin(\alpha)\cos(\alpha)\\ 2\sin(\alpha)\cos(\alpha)&=\left(\sin(\alpha)+\cos(\alpha)\right)^2\\ &-\sin^2(\alpha)-\cos^2(\alpha)\\ 2\sin(\alpha)\cos(\alpha)&=\left(\sin(\alpha)+\cos(\alpha)\right)^2-1 \end{split}
  • If we also use the definition of the tangent as the ratio between sine and cosine, we can find a third way to express sin(2α)\sin(2\alpha):
2sin(α)cos(α)=2(tan(α)cos(α))cos(α)=2tan(α)cos2(α)=2tan(α)1cos2(α)=2tan(α)cos2(α)+sin2(α)cos2(α)=2tan(α)1+tan2(α)\scriptsize\quad\begin{split} 2\sin(\alpha)\cos(\alpha)&=2\left(\tan(\alpha)\cos(\alpha)\right)\cos(\alpha)\\ &=2\tan(\alpha)\cos^2(\alpha)\\[1em] &=\frac{2\tan(\alpha)}{\frac{1}{\cos^2(\alpha)}}\\[1em] &=\frac{2\tan(\alpha)}{\frac{\cos^2(\alpha)+\sin^2(\alpha)}{\cos^2(\alpha)}}\\[1em] &=\frac{2\tan(\alpha)}{1+\tan^2(\alpha)}\\[1em] \end{split}

Calculate the double angle identity for the cosine

If the double angle identity for the sine uses the product of sine and cosine, the one for the cosine uses the difference of these functions:

cos(2α)=cos2(α)sin2(α)\scriptsize \cos(2\alpha) =\cos^2(\alpha) - \sin^2(\alpha)

Again, we can find other ways to express this identity. The first one uses the Pythagorean identity:

cos(2α)=cos2(α)sin2(α)=cos2(α)(1cos2(α))=2cos2(α)1=12sin2(α)\scriptsize\begin{split} \cos(2\alpha) &=\cos^2(\alpha) - \sin^2(\alpha)\\ &=\cos^2(\alpha) - \left(1-\cos^2(\alpha)\right)\\ &=2\cos^2(\alpha)-1\\ &=1-2\sin^2(\alpha) \end{split}

Introducing the tangent once again gives us the second form of the double angle identity for the cosine:

cos(2α)=2cos2(α)1=11+tan2(α)1=1tan2(α)1+tan2(α)\scriptsize\begin{split} \cos(2\alpha) &=2\cos^2(\alpha) -1\\ &=\frac{1}{1+\tan^2(\alpha)}-1\\[1em] &=\frac{1-\tan^2(\alpha)}{1+\tan^2(\alpha)} \end{split}

Double angle trig identity for the tangent

To calculate the double angle formula for the tangent, we can use the ratio between the results found previously for the sine and cosine. There is a single neat way to express this identity:

tan(2α)=2tan(α)1tan2(α)\scriptsize \tan(2\alpha) = \frac{2\tan(\alpha)}{1-\tan^2(\alpha)}

If our trig double angle identities calculator was helpful, we suggest you visit our other tools related to the topic:

FAQ

How do I find the double angle identity for the sine?

To find the double angle trig identity for the sine, follow these easy steps:

  1. Start with the compound angle formula for the sine: sin(α + ß) = sin(α)cos(ß) + sin(ß)cos(α).
  2. Substitute the angle ß with α,
  3. The result is the following formula: sin(α + α) = sin(2α) = sin(α)cos(α) + sin(α)cos(α) = 2sin(α)cos(α).

What are the double angle identities in trigonometry?

The double angle trig identities are a set of trigonometric identities that allow you to compute the values of the trigonometric functions of angles in the form when the value of sin(α), cos(α), or tan(α) is known. Here are the identities for the three fundamental trigonometric functions:

  • sin(2α) = 2sin(α)cos(α);
  • cos(2α) = cos²(α) - sin²(α); and
  • tan(2α) = 2tan(α)/(1 - tan²(α)).

How do I find the cosine of 120 degrees?

To find the cosine of 120 degrees, you can use the double angle identity of trigonometry for the cosine. Use the following formula:
cos(2α) = cos²(α) - 1
If you know the cosine for this one:
cos(2α) = 1 + sin²(α)
If you know the sine.
Choosing 2α = 120°, and knowing that sin(60) = sqrt(3)/2, write:
cos(120°) = 1 - sin²(60°) = 1 - 3/4 = 2/4 = 1/2

Davide Borchia
Angle θ
deg
Step by step solution?
yes, please
Double sine
Double cosine
Double tangent
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