Half Angle Formula

Trigonometry is one of the important branches in the domain of mathematics. This concept was given by the Greek mathematician Hipparchus. In this topic, we will see the concept of trigonometric ratios with half-angle values and also half angle formula with examples. Let us learn it!

Half Angle Formula

Concept of Trigonometry:

Actually, trigonometry is the study of triangles where we deal with the angles as well sides of the triangle.  Specifically, it is all about a right-angled triangle. It is one of those parts of mathematics which helps in finding the angles and missing sides of a triangle, it takes the help of the trigonometric ratios. Various formulas are there with complete angles as well as half angles

In a right-angled triangle, the angles are either measured in radians or degrees. The common trigonometry angles are 0°, 30°, 45°, 60°, and 90°. Various formulas are there to use these angles and provide sides easily. But also we may have a formula for half-angle values, which are commonly used.

This branch of mathematics divides into two sub-branches called plane trigonometry and spherical geometry. The trigonometric ratios of a triangle are the trigonometric functions. Sine, cosine, and tangent are 3 important and heavily used trigonometric functions. Let us take a right-angled triangle, in which the longest side is the hypotenuse. And the sides opposite to the hypotenuse is referred to as the adjacent and opposite.

The trigonometric ratios like sine, cosine, and tangent of given angles are easy to memorize. We will also show the table where all the ratios and angles are given. To find values of other angles we may use trigonometric functions and formulas.

For example, in a right-angled triangle,

• \(Sin  \theta = \frac {Perpendicular} {Hypotenuse}\)
• i.e. \(Sin \theta = \frac{P}{H}\)
• or \(\theta = sin^{-1} \frac{P}{H}\)

Similarly,

• \(\theta  = cos ^{-1} \frac{B}{H}\)
• \(\theta = tan ^{-1} \frac{P}{B}\)

Source: en.wikipedia.org

Some Half Angle Formula

The Trigonometric formulas or Identities are the equations which are used extensively in many problems of mathematics as well as science. Some of the popular half angle formulas are given as below:

1. Sin half-angled formula: \(sin \;\frac{ \theta }{2} = \sqrt( \frac{(1 – cos \theta) }{2})\)
2. Cos half-angled formula: \(cos \frac{\theta}{2} =  \sqrt(\frac{1 + cos \theta}{2})\)
3. The tan half-angled formula:
   • \(tan \frac {\theta}{2} = \frac {1-cos \theta }{sin \theta}\)
   • \(tan \frac {\theta}{2} = \frac {sin \theta}{1+ cos \theta }\)

Solved Examples for Half Angle Formula

Q.1: Find the value of Sin 30 degrees by using the sine half-angle formula.

Solution: Given angle \(\theta = 60 degrees.\)

Now using the sine half angle formula as given,

\(sin \frac{ \theta }{2} = \sqrt( \frac{(1 – cos \theta) }{2})\)

substituting the values of \theta we get,

\(sin \frac{60}{2} =  \sqrt(\frac{1 – cos\; 60}{ 2})\)

we know that \(cos 60 = \frac {1}{2} = 0.5 then,\)

\(sin 30 = \pm \sqrt (\frac{1- 0.5}{2})\)

\(sin \; 30 = 0.5\)

Therefore value of sin 30 will be\( \frac{1}{2}\)

Q.2: Find the value of tan 30 degrees by using the tan half-angle formula.

Solution:

Given angle \(\theta = 60 degrees.\)

Now using the tan half angle formula as given,

tan \(\frac {\theta}{2} = \frac {1-cos \theta }{sin \theta}\)

substituting the values of \(\theta\) we get,

\(tan \frac {60}{2} = \frac {1-cos\; 60 }{sin\; 60}\)

we know that \(cos 60 = \frac {1}{2}\)

and \(sin 60 = \frac{\sqrt 3}{2}\) then, we have

\(tan \frac {60}{2} = \frac {1-cos\; 60 }{sin\; 60}\)

\(tan \frac {60}{2} = \frac {1-\frac{1}{2}}{\frac{\sqrt 3}{2} }\)

\(tan\; 30 = \frac {\frac{1}{2}}{\frac{\sqrt 3}{2} }\)

\(tan\; 30 = \frac {1}{\sqrt 3}\)

Therefore value of \(tan\;30 = \frac {1}{\sqrt 3}\)