Half-Angle Identities

Related Topics

• How to Solve Double Angle Identities?

Step-by-step guide to half-angle identities

The half angle formulas are derived from double angle formulas and are expressed in terms of half angles such as \(\frac{\theta }{2}, \frac{x}{2}, \frac{A}{2}\). Half-angle formulas are used to find the exact values of trigonometric ratios of angles such as \(22.5°\) (which is half of the standard angle of \(45°\)), \(15°\) (which is half of the standard angle of \(30°\)).

Here is a list of important half-angle formulas:

• \(\color{blue}{sin\:\left(\frac{A}{2}\right)=\pm \sqrt{\frac{1-\:cos\:A}{2}}}\)
• \(\color{blue}{cos\:\left(\frac{A}{2}\right)=\pm \sqrt{\frac{1+\:cos\:A}{2}}}\)
• \(\color{blue}{tan\:\left(\frac{A}{2}\right)=\pm \sqrt{\frac{1-\:cos\:A}{1+cos\:A}}=\frac{sin\:A}{1+cos\:A}=\frac{1-cos\:A}{sin\:A}}\)

Half-Angle Identities – Example 1:

Find \(sin\:15^{\circ }\) using a half-angle formula.

Solution:

Since \(15^{\circ }=\frac{30^{\circ }}{2}\), we can use this formula: \(sin\:\left(\frac{A}{2}\right)=\pm \sqrt{\frac{1-\:cos\:A}{2}}\)

\(sin \frac{30^{\circ }}{2}=\sqrt{\frac{1-\:cos\:30^{\circ }}{2}}\)

\(=\sqrt{\frac{1-\sqrt{\frac{3}{2}}}{2}}\)

\(=\sqrt{\frac{\frac{2-\sqrt{3}}{2}}{2}}\)

\(=\sqrt{\frac{2-\sqrt{3}}{4}}\)

\(=\frac{\sqrt{2-\sqrt{3}}}{2}\)

we used the positive root because \(sin\:15^{\circ }\) is positive.

Exercises for Half-Angle Identities

Find each value using a half-angle formula.

1. \(\color{blue}{sin\:165^{\circ }}\)
2. \(\color{blue}{tan\:\frac{\pi }{8}}\)
3. \(\color{blue}{sin\:67.5^{\circ }}\)
4. \(\color{blue}{tan\:\frac{7\pi }{8}}\)

1. \(\color{blue}{\frac{\sqrt{6}-\sqrt{2}}{4}}\)
2. \(\color{blue}{\sqrt{2}-1}\)
3. \(\color{blue}{\frac{\sqrt{2+\sqrt{2}}}{2}}\)
4. \(\color{blue}{1-\sqrt{2}}\)