Sine Half Angle Formula

Trigonometry is a field of mathematics that explores the connections between right triangle side lengths and angles. It’s the study of how a right-angled triangle’s sides and angles are connected. The trigonometric ratios that are used to study this field of mathematics are sine, cosine, tangent, cotangent, secant, and cosecant. The terms ‘Trigonon’ and ‘Metron,’ which symbolize a triangle and a measurement, respectively, are combined to make trigonometry. It, therefore, aids in the calculation of unknown dimensions of a right-angled triangle by employing equations and identities based on this relationship.

Sine Trigonometric Ratio

A trigonometric ratio is defined as the ratio of any two right triangle sides. The 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.

sin θ/2 = ±√((1 – cos θ) / 2)

Derivation

The 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.

Sample Problems

Problem 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

Problem 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)

Problem 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

Problem 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

Problem 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

Problem 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))

Problem 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