Sine half angle is calculated using various formulas and there are multiple ways to prove the same. In this article, we have covered formulas related to the sine half angle, its derivation-related examples and others in detail.

Sine Trigonometric Ratio

Sine ratio is expressed as the ratio of the opposing side’s length divided by the hypotenuse’s length. It is denoted by the abbreviation sin.

If θ is the angle that lies between the base and hypotenuse of a right-angled triangle then,

sin θ = Perpendicular/Hypotenuse

Sine Half Angle (Sin θ/2) Formula

Half-angle formulae are generally expressed by θ/2 in trigonometry, where θ is the angle. The half-angle is a sub-multiple angle in this case. The half-angle formulae are used to calculate the precise values of trigonometric ratios of standard angles like 30°, 45°, and 60°.

Using the ratio values for these conventional angles, we can obtain the ratio values for difficult angles like 22.5° (half of 45°) or 15° (half of 30°). The sine half-angle is denoted by the abbreviation sin θ/2. It is a trigonometric function that returns the sin function value for a half-angle. The function sin θ has a period of 2 while sin θ/2 has a period of 4.

Formula for sin(θ/2) is represented below:

Sine Half Angle Formula

Derivation of Sine Half Angle Formula

Formula for sine half angle is derived by using the double angle formulas for sine and cosine.

We know, cos 2θ = 1 – 2 sin² θ  …… (1)

Substitute θ as θ/2 in the equation (1)

=> cos θ = 1 – 2 sin² (θ/2)

Solve the equation for sin θ/2

=> 2 sin² (θ/2) = 1 – cos θ

=> sin² (θ/2) = (1 – cos θ)/2

=> sin θ/2 = ±√((1 – cos θ) / 2)

This derives the formula for sine half angle ratio.

Read More:

• Sine Half Angle Formula
• Double Angle Formulas
• Trigonometry Formulas

Examples of Sine Half-Angle Formula

Example 1. If cos θ = 3/5, find the value of sin θ/2 using the half-angle formula.

Solution:

We have, cos θ = 3/5

Using the formula we get,

sin θ/2 = √((1 – cos θ) / 2)

= √((1 – 3/5) / 2)

= √((2/5) / 2)

= 1/√5

Example 2. If cos θ = 12/13, find the value of sin θ/2 using the half-angle formula.

Solution:

We have, cos θ = 12/13

Using the formula we get,

sin θ/2 = √((1 – cos θ) / 2)

= √((1 – 12/13) / 2)

= √((1/13) / 2)

= √(2/5)

Example 3. If sin θ = 8/17, find the value of sin θ/2 using the half-angle formula.

Solution:

We have, sin θ = 8/17

Find the value of cos θ using the formula sin² θ + cos² θ = 1.

cos θ = √(1 – (64/289))

= √(225/289)

= 15/17

Using the formula we get,

sin θ/2 = √((1 – cos θ) / 2)

= √((1 – 15/17) / 2)

= √((2/17) / 2)

= 1/√17

Example 4. If sec θ = 5/4, find the value of sin θ/2 using the half-angle formula.

Solution:

We have, sec θ = 5/4.

Using cos θ = 1/sec θ, we get cos θ = 4/5.

Using the formula we get,

sin θ/2 = √((1 – cos θ) / 2)

= √((1 – 4/5) / 2)

= √((1/5) / 2)

= 1/√10

Example 5. If tan θ = 12/5, find the value of sin θ/2 using the half-angle formula.

Solution:

We have, tan θ = 12/5.

Clearly, cos θ = 5/√(12² + 5²) = 5/13

Using the formula we get,

sin θ/2 = √((1 – cos θ) / 2)

= √((1 – 5/13) / 2)

= √((8/13) / 2)

= √(4/13)

= 2/√13

Example 6. If cot θ = 8/15, find the value of sin θ/2 using the half-angle formula.

Solution:

We have, cot θ = 8/15.

Clearly, cos θ = 8/√(8² + 15²) = 8/17

Using the formula we get,

sin θ/2 = √((1 – cos θ) / 2)

= √((1 – 8/17) / 2)

= √((9/17) / 2)

= 3(√(2/17))

Example 7. Find the value of sin 15° using the half-angle formula.

Solution:

We have to find the value of sin 15°.

Let us take θ/2 = 15°

=> θ = 30°

Using the half angle formula we have,

sin θ/2 = √((1 – cos θ) / 2)

= √((1 – cos 30°) / 2)

= √((1 – (√3/2)) / 2)

= (2 – √3)/4

Practice Problems on Sine Half Angle Formula

1. Find sin(15°) using the sine half-angle formula.

2. Calculate sin(π/8) using the sine half-angle formula.

3. Express sin(22.5°) in terms of √2 using the sine half-angle formula.

4. If cos(θ) = 3/5, find sin(θ/2) using the sine half-angle formula.

5. Prove that sin(π/12) = (√6 – √2) / 4 using the sine half-angle formula.

6. Find the exact value of sin(7.5°) using the sine half-angle formula.

7. If tan(θ) = 4/3, calculate sin(θ/2) using the sine half-angle formula.

8. Express sin(π/16) in terms of √2 using the sine half-angle formula.

9. Find sin(θ/2) if cos(θ) = -0.6, using the sine half-angle formula.

10. Prove that sin(11.25°) = (√2 – 1) / (2√2) using the sine half-angle formula.

Summary

The sine half-angle formula, expressed as sin(θ/2) = ±√((1 – cos(θ))/2), is a fundamental tool in trigonometry used to calculate the sine of half an angle when the cosine of the full angle is known. Derived from the cosine double angle formula, it’s particularly useful for dealing with angles that are fractions of standard angles. This formula often results in simplified radical expressions and is crucial for solving complex trigonometric problems. Its applications extend to various fields of mathematics and engineering, making it an essential concept to master. The formula’s structure, involving a square root, allows for precise calculations and helps in simplifying otherwise complex trigonometric expressions. Understanding and applying this formula enhances one’s problem-solving skills in trigonometry and related areas of mathematics.

FAQs on Sine Half Angle Formula

What is the formula for the half-angle formula sina.sinb?

Formula for sin(a).sin(b) is, sin(a).sin(b) = (1/2)[cos(a – b) – cos(a + b)].

What is the formula for sin 45 half angle?

Formula for sin 45° in terms of half angle is ±√((1 – cos 90°) / 2) and its value is 1/√2.

What is the formula of sin 120 half angle?

Formula for sin 120° in terms of half angle is ±√((1 – cos 240°) / 2) and its value is √3/2.

What is half of sin?

Half angle formula of sin is, sin A/2 = ±√[(1 – cos A) / 2]