Half Angle formula

In trigonometry, the half-angle formula is used to determine the exact values of the trigonometric ratios of angles such as 15° (half of the standard angle 30°), 22.5° (half of the standard angle 45°), and so on. If θ is an angle, then the half angle is represented by θ/2. We know that the trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent. With the help of the trigonometry table, we will get to know the values for the standard angles like 0°, 30°, 45°, 60°, 90° for these functions. But if we want to know the trigonometric values for the angles like 15°, 22.5°, and so on, the half angle formula is extremely useful. Now, let us discuss the half angle formula for the three primary trigonometric functions.

Half Angle Formulas

Generally, the half angle formulas are derived from the double angle formulas and the double angle formulas are derived from the angle sum and angle difference formulas in trigonometry.

Half Angle Formula for Sine Function

\(\begin{array}{l}sin \frac{a}{2} = \pm \sqrt{\frac{(1- \cos a)}{2}}\end{array} \)

Half Angle Formula for Cosine Function

\(\begin{array}{l}cos \frac{a}{2} = \pm \sqrt{\frac{(1+ \cos a)}{2}}\end{array} \)

Half Angle Formula for Tangent Function

\(\begin{array}{l}tan \left ( \frac{a}{2} \right ) = \frac{1 – \cos a}{\sin a} = \frac{\sin a}{1 + \cos a}\end{array} \)

Solved Example on Half Angle Formula

Example:

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

Solution:

We know that

\(\begin{array}{l}sin \frac{a}{2} = \pm \sqrt{\frac{1-cos a}{2}}\end{array} \)

Now, substitute a = 30° on both sides, we get

\(\begin{array}{l}sin \frac{30^{\circ}}{2} = \pm \sqrt{\frac{1-cos 30^{\circ}}{2}}\end{array} \)

\(\begin{array}{l}sin 15^{\circ} = \pm \sqrt{\frac{1-cos 30^{\circ}}{2}}\end{array} \)

We know that cos 30° =√3/2 , and substitute it in the above equation,

\(\begin{array}{l}sin 15^{\circ} = \pm \sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}}\end{array} \)

\(\begin{array}{l}sin 15^{\circ} = \pm \sqrt{\frac{2-\sqrt{3}}{4}}\end{array} \)

\(\begin{array}{l}sin 15^{\circ} = \pm \frac{\sqrt{2-\sqrt{3}}}{2}\end{array} \)

Now, substitute √3 = 1.732, we get

\(\begin{array}{l}sin 15^{\circ} = \pm \frac{\sqrt{2-1.732}}{2}\end{array} \)

\(\begin{array}{l}sin 15^{\circ} = \pm \frac{0.517687}{2}\end{array} \)

\(\begin{array}{l}sin 15^{\circ} = \pm 0.2588\end{array} \)

Since sin 15° lies in the first quadrant, the value of sin 15° should be positive. 

Therefore, the value of sin 15° is 0. 2588.

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Frequently Asked Questions on Half Angle Formula

What is the half angle formula?

In trigonometry, a half angle formula is used to determine the exact values for the half angles like 15°, 22.5°, and so on.

What is the half angle formula for sine function?

The half angle formula for sine function is sin a/2 = ±√[(1 – cos a) / 2]

What is the half angle formula for cosine function?

The half angle formula for cosine function is cos a/2 = ±√[(1 + cos a) / 2]

What is the half angle formula for tangent function?

The half angle formula for tangent function is tan a/2 = ±√[1 – cos a] / [1 + cos a]

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