Half-angle formulas are used to find various values of trigonometric angles, such as for 15°, 75°, and others, they are also used to solve various trigonometric problems.
Several trigonometric ratios and identities help in solving problems of trigonometry. The values of trigonometric angles 0°, 30°, 45°, 60°, 90°, and 180° for sin, cos, tan, cosec, sec, and cot are determined using a trigonometry table.
Half-angle formulas are trigonometric identities that express the sine, cosine, and tangent of half an angle (θ/2) in terms of the sine or cosine of the full angle θ. They are especially useful in simplifying expressions and solving trigonometric equations.
Half Angle Formulas Derivation Using Double Angle Formulas
Half-angle formulas are derived using double-angle formulas. Before learning about half-angle formulas, we must learn about Double-angle in Trigonometry, The most commonly used double-angle formulas in trigonometry are:
- sin 2x = 2 sin x cos x
- cos 2x = cos2 x - sin2 x
= 1 - 2 sin2x
= 2 cos2x - 1 - tan 2x = 2 tan x / (1 - tan2x)
Now replacing x with x/2 on both sides in the above formulas we get
- sin x = 2 sin(x/2) cos(x/2)
- cos x = cos2 (x/2) - sin2 (x/2)
= 1 - 2 sin2 (x/2)
= 2 cos2(x/2) - 1 - tan A = 2 tan (x/2) / [1 - tan2(x/2)]
Read More: Double Angled Formulas
Half-Angle Formula for Cos Derivation
We use cos2x = 2cos2x - 1 to find the Half-Angle Formula for Cos
Put x = 2y in the above formula
cos (2)(y/2) = 2cos2(y/2) - 1
cos y = 2cos2(y/2) - 1
1 + cos y = 2cos2(y/2)
2cos2(y/2) = 1 + cosy
cos2(y/2) = (1+ cosy)/2
cos(y/2) = ± √{(1+ cosy)/2}
Half-Angle Formula for Sin Derivation
We use cos 2x = 1 - 2sin2x for finding the Half-Angle Formula for Sin
Put x = 2y in the above formula
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}
Half-Angle Formula for Tan Derivation
We know that tan x = sin x / cos x such that,
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)
Also Check
Solved Examples of Half Angle Formulas
Example 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
Example 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
Example 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
Example 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
Example 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
Example 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