Double Angle Formula

Trigonometry the study of the relationships which involve angles, lengths, and heights of triangles.  It has many useful identities for learning and deriving the many equations and formulas in science. This article looks at some specific kinds of trigonometric formulae which are popular as the double angle formulae. They are known as this because they involve trigonometric functions of double angles. We shall learn the double angle formula with examples. Let us begin!

What is a Double Angle?

Double angle identities and formula are useful for solving certain integration problems where a double formula may make things much simpler to solve. Thus in math as well as in physics, these formulae are useful to derive many important identities. Notice that there are many listings for the double angle for sine, cosine, and tangent.

The Double Angle Formula

We may start by recalling the addition formulae of trigonometry ratios with two angles A and B. Students, are already knowing these.

• \(sin(X + Y) = sin X cos Y + cos X sin Y\)
• \(cos(X + Y) = cos X cos Y − sin X sin Y\)
• \(tan(X + Y) =  \frac {tan X + tan Y}{ 1 − tan X tan Y}\)

We may consider what happens if we let Y equal to X. Then the first of these formulae will become: \(sin (X + X) = sin X cos X + cos X sin X\)

So, \(sin 2X = 2 sin X cos X\)

This is the first double-angle formula. Here we are using the doubling of the angle (as in 2X). Now, if we put Y equal to X in the second addition formula then we have,

\(cos(X + X) = cos X cos X − sin X sin X\)

i.e.  \(cos 2X = cos^{2} X – sin ^{2} X\)

And this is our second double angle formula.

Similarly,

\(tan(X + X) = \frac {tan X + tan X} { 1 − tan X tan X}\)

So that, \(tan 2X = \frac {2 tan X} { 1 – tan^{2} X}\)

These three double angle formulae are popular as well as much useful.

Solved Examples for Double Angle Formula

Q.1: If \(sin x =frac {3}{5}\) . Then find cos 2x.

Solution: We have

\(Cos 2x=1- 2 sin ^{2}x\\\)

\(cos 2x = 1–2(\frac{3}{5}) ^{2}\\\)

\(cos 2x = \frac {7}{25}\)

Thus \(cos 2x =  \frac {7}{25}\)

Example-2: Solve the equation given as: \(sin 2x = sin x , 0\leq x < \pi\)

Solution: In this case we will use the double angle formulae as \(sin 2x = 2 sin x cos x.\)

This will give, \(2 sin x cos x = sin x\)

We will rearrange this and factorise as follows: \(2 sin x cos x − sin x = 0\)

\(sin x (2 cos x − 1) = sin x = 0\)

or \((2 cos x – 1) = 0\)

We will deal the first one with sin x = 0. Which will give x = 0, in the given interval.

Also, the equation \(2 cos x − 1 = 0\) will give

\(cos x = \frac {1} {2}\)

i.e.\( x = \frac { \pi}{3}\)

Thus solutions of the equation are 0 and \(\frac{\pi}{3}\)