Double Angle Formulae

Introduction

The study of trigonometry involves the study of relationships between angles and sides of triangles. While working with triangles, although we generally work with a single angle, say θ, sometimes we also use double angles or 2θ. 

These double angle formulae are used to express double angles (2θ) trigonometric ratios in terms of single angle (θ) trigonometric ratios. 

These double angles are special cases and are derived from the trigonometry sum formulas.

For example, the value of sin30 can be used to find the value of sin 60. The double angle formulae may also be used to derive the triple angle (3θ) formulae.

Double Angle Formulae

Double angle formulae and identities can be used to solve complex and higher-level integration problems. These double formulae make working out complex calculations much more manageable. These formulae can also be used to derive many vital identities in both math and physics. 

Double angle formulae use the single angle values of trigonometric functions to find the values of double angles. 

They are derived from the trigonometric sum of two angles formulae.

Let us consider sum formulae in trigonometry.

1. sin (X + Y) = sin X cos Y + cos X sin Y
2. cos (X + Y) = cos X cos Y – sin X sin Y
3. tan (X + Y) = (tan X + tan Y) / (1 – tan X tan Y)

In the sum formula, two different angles, X and Y, are added. However, for double angles, a substitution of X = Y takes place in each formula. This will give the formula for 2X or double angle X for sin, cos, and tan functions. 

Pythagorean identities can also be used to derive alternative formulae in this context.

The double angle formulas for sin, cos, and tan are as follows:

1. sin 2X = 2 sin X cos X OR 

sin2X= (2 tan X) / (1 + tan2X)

1. cos 2X = cos2X – sin2X OR 

cos 2X= 2cos2X – 1 OR 

cos 2X= 1 – 2sin2X OR 

cos 2X= (1 – tan2X) / (1 + tan2X)

1. tan 2X = (2 tan X) / (1 – tan2X)

The Derivation of Double Angle Formulae

Double angle formulae are derived in the following ways for sin, cos, and tan. 

Double Angle Formula Derivation for Sin

The sum formula for the sin function is given as follows: 

sin (X + Y) = sin X cos Y + cos X sin Y

Here, if the substitution X=Y is implemented, the above formula becomes:

sin (X+X) = sin X cos X + cos X sin X

sin 2X = 2 sin X cos X

Another formula for sin 2A can also be derived using tan A.

To do so, the identity sec2 X = 1 + tan2 X will be used.

As derived above:

sin 2X= 2 sinX cosX

Multiplying above and below by cos X,

sin2X= 2sinXcosX ∙ cos2 X

sin2X= 2tanX. cos2 X

Now cos2X can be written as 1/ sec2 X

So, sin 2X = 2 tan X ∙ 1/ sec2 X

Using the identity, sec2 X = 1 + tan2X, we can derive:

sin 2X = 2tanX/ (1+tan2 X)

Either of the previously derived double angle formulae may be used for sine function:

sin 2X = 2 sinX cos X 

OR

sin2X= (2 tan X) / (1 + tan2 X)

Double Angle Formula Derivation for Cos

For this derivation, the sum formula for the cosine function will be used:

cos(X+Y) = cos X cos Y – sin X sin Y

If the substitution Y=X is implemented, the above formula becomes,

cos (X+X) = cos X cos X – sin X sin X

cos2X = cos2 X – sin2 X

Consider the Pythagorean identity:

sin2 X + cos2 X = 1

This identity may be used to derive another version of cos2X.

(i) If sin2 X=1-cos2 X is substituted in in the cos2X formula:

cos2X = cos2 X − (1 − cos2 X) 

OR

cos2X= 2cos2 X – 1

(ii) If cos2 X= 1- sin2 X is substituted in the cos2X formula

cos2X = (1- sin2 X) – sin2 X 

OR

cos2X= 1 – 2sin2 

The formula of cos 2X may also be derived in terms of tan using the base formula.

cos2X=cos2X−sin2X

cos2X=cos2X (1−sin2X cos2X)

cos 2X = cos2 X-sin2 X

Multiplying above and below by cos2X,

cos 2X = (cos2X−sin2X)/cos2X.cos2X

cos 2X = cos2 X (1 – tan2 X)

Since cos2X can be written as 1/ sec2X

So: cos 2X = (1 – tan2 X)/ sec2X

Using the identity sec2X = 1 + tan2X,

cos 2X = (1−tan2X)/ (1+tan2X)

Thus, the double angle formulas of the cosine function are:

cos2X = cos2X – sin2X

OR

cos2X= 2cos2X – 1

OR 

cos2X= 1 – 2sin2X

OR

cos2X= (1 – tan2X) / (1 + tan2X)

Double Angle Formula Derivation for Tan

The sum formula for tan function may be expressed as:

tan (X + B) = (tan X + tan B) / (1 – tan X tan B)

If the substitution B=X is implemented to find the double angle of X,

tan (X + X) = (tan X + tan X) / (1 – tan X tan X) 

OR 

tan2X= (2 tan X) / (1 – tan2X)

Area of a Right Triangle

We know that area of a right-angled triangle with a given base b and height h is:

area=12bh by the right triangle formula.

Consider the triangle ABC given below, right-angled at D.

From this figure, the area of the triangle can be written as ½ a* height (by right triangle formula).

The height of the triangle above is b sin C.

So, the area of a right triangle can also be written as:

area=12ab sin C=ab sin C2

Conclusion

Complex trigonometric expressions may be simplified using double-angle formulas. As a result, it is very useful to know how to derive these formulae; being able to apply any of the various expressions of these formulae is invaluable as they can be adapted to specific uses as required in a given problem. The formulae for triple angles and other higher angles may also be similarly derived by expressing them in terms of double angles.