Solved Examples on 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

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 widely used in mathematics, let’s learn about them in detail in this article.