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Half Angle Formulas

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  • Last Updated : 03 Mar, 2022

Trigonometry is the branch of mathematics that deals with angles, dimensions and measures. An Angle is formed when the two lines are at some inclination with respect to each other. The inclination of angle is called measurement and when several angles come together, they give rise to a dimension.

There are several formulas and identities that help in determining the inclination and measurements. The values of trigonometric 0°, 30°, 45°, 60°, 90°, 180° for sin, cos, tan, cosec, sec and cot are determined. Similarly, we also have something called the half-angle formula in mathematics. 

Half-Angle Formulae

For finding the values of angles apart from the well-known values of 0°, 30°, 45°, 60°, 90°, 180°. Half angles are derived from double angle formulas and are listed as below for sin, cos, and tan:

  1. sin x/2 = +/- ((1 – cos x)/ 2)1/2
  2. cos x/2 = +/- ((1 + cos x)/ 2)1/2
  3. tan x/ 2 = (1 – cosx)/ sinx

Some more important identities of double angle formula which will be useful for the derivation of half-angle formulas,

cos 2x = cos (x +x) 

cos 2x = (cosx)(cosx) – (sinx)(sinx)

cos 2x = cos2x – sin2x

cos 2x = cos2x – (1 -cos2x)

cos2x = 2cos2x – 1 ⇢ (eq. 1)

Now, put cos2x = 1 – sin2x

cos2x = 2 (1 – sin2x) – 1 

cos2x = 2 – 2sin2x- 1 

cos2x = 1 – 2sin2x ⇢ (eq. 2)

Derivation of half-angle formula for cos

From above, we will make use of cos2x = 2cos2x – 1, equation 1 denoted by eq. 1,

Put x = 2y

cos (2)(y/2) = 2cos2(y/2) – 1

cos y = 2cos2(y/2) – 1

1 + cos y = 2cos2(y/2) 

Or 

2cos2(y/2) = 1 + cosy

cos2(y/2) = (1+ cosy)/2

cos(y/2) = +/- √(1+ cosy)/2

Derivation of half-angle formula for sin

From above, we will make use of cos2x = 1 – 2sin2x , equation 2 denoted by eq2.

Put x = 2y

cos (2)(y/2) = 1 – 2sin2(y/2)     

cos y = 1 – 2sin2(y/2)   

2sin2(y/2) = 1 – cosy

sin2(y/2) = (1 – cosy)/2

sin(y/2) = +/- √(1 – cosy)/2

Derivation of half-angle formula for tan

tan(x/2) = sin(x/2) / cos(x/2)

Putting the values of half angle for sin and cos. We get,

tan(x/2) = +/- ((√(1 – cosy)/2 ) / (√(1+ cosy)/2 ))

tan(x/2) = +/- (√(1 – cosy)/(1+ cosy) )

Rationalising the denominator

tan(x/2) = +/- (√(1 – cosy)(1 – cosy)/(1+ cosy)(1 – cosy))

tan(x/2) = +/- (√(1 – cosy)2/(1 – cos2y))

tan(x/2) = +/- (√(1 – cosy)2/( sin2y))

tan(x/2) = (1 – cosy)/( siny)

Sample Problems

Question 1: Determine the value of sin 15°

Solution:

We know that the formula for half angle of sine is given by:

sin x/2 = +/- ((1 – cos x)/ 2) 1/2

The value of sine 15° can be found by substituting x as 30° in the above formula

sin 30°/2 = +/- ((1 – cos 30°)/ 2) 1/2

sin 15° = +/- ((1 – 0.866)/ 2) 1/2

sin 15° = +/- (0.134/ 2) 1/2

sin 15° = +/- (0.067) 1/2

sin 15° = +/- 0.2588

Question 2: Determine the value of sin 22.5°

Solution:

We know that the formula for half angle of sine is given by:

sin x/2 = +/- ((1 – cos x)/ 2) 1/2

The value of sine 15° can be found by substituting x as 45° in the above formula

sin 45°/2 = +/- ((1 – cos 45°)/ 2) 1/2

sin 22.5° = +/- ((1 – 0.707)/ 2) 1/2

sin 22.5° = +/- (0.293/ 2) 1/2

sin 22.5° = +/- (0.146) 1/2

sin 22.5° = +/- 0.382

Question 3: Determine the value of tan 15°

Solution:

We know that the formula for half angle of sine is given by:

tan x/2 = +/- (1 – cos x)/ sin x

The value of tan 15° can be found by substituting x as 30° in the above formula

tan 30°/2 = +/- (1 – cos 30°)/ sin 30°

tan 15° = +/- (1 – 0.866)/ sin 30

tan 15° = +/- (0.134)/ 0.5

tan 15° = +/- 0.268

Question 4: Determine the value of tan 22.5°

Solution:

We know that the formula for half angle of sine is given by:

tan x/2 = +/- (1 – cos x)/ sin x

The value of tan 22.5° can be found by substituting x as 45° in the above formula

tan 30°/2 = +/- (1 – cos 45°)/ sin 45°

tan 22.5° = +/- (1 – 0.707)/ sin 45°

tan 22.5° = +/- (0.293)/ 0.707

tan 22.5° = +/- 0.414

Question 5: Determine the value of cos 15°

Solution:

We know that the formula for half angle of sine is given by:

cos x/2 = +/- ((1 + cos x)/ 2) 1/2

The value of sine 15° can be found by substituting x as 30° in the above formula

cos 30°/2 = +/- ((1 + cos 30°)/ 2) 1/2

cos 15° = +/- ((1 + 0.866)/ 2) 1/2

cos 15° = +/- (1.866/ 2) 1/2

cos 15° = +/- (0.933) 1/2

cos 15° = +/- 0.965

Question 6: Determine the value of cos 22.5°

Solution:

We know that the formula for half angle of sine is given by:

cos x/2 = +/- ((1 + cos x)/ 2) 1/2

The value of sine 15° can be found by substituting x as 45° in the above formula

cos 45°/2 = +/- ((1 + cos 45°)/ 2) 1/2

cos 22.5° = +/- ((1 + 0.707)/ 2) 1/2

cos 22.5° = +/- (1.707/ 2) 1/2

cos 22.5° = +/- ( 0.853 ) 1/2

cos 22.5° = +/- 0.923

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