Ratios & Identities: Trigonometric Ratios of Multiple Angles

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. The trigonometric ratios of multiple angles are an extension of the basic trigonometric functions to angles that are multiples of a given angle. These ratios are crucial in various fields, including physics, engineering, and mathematics.

Basic Trigonometric Functions

Before diving into multiple angles, let's review the basic trigonometric functions for an angle $\theta$:

• Sine ($\sin \theta$)
• Cosine ($\cos \theta$)
• Tangent ($\tan \theta$)

These functions are defined for a right-angled triangle as follows:

• $\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}$
• $\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}$
• $\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{\sin \theta}{\cos \theta}$

Trigonometric Ratios of Double Angles

When dealing with double angles, we use the following identities:

• $\sin(2\theta) = 2\sin \theta \cos \theta$
• $\cos(2\theta) = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta$
• $\tan(2\theta) = \frac{2\tan \theta}{1 - \tan^2 \theta}$

Trigonometric Ratios of Triple Angles

For triple angles, the identities become more complex:

• $\sin(3\theta) = 3\sin \theta - 4\sin^3 \theta$
• $\cos(3\theta) = 4\cos^3 \theta - 3\cos \theta$
• $\tan(3\theta) = \frac{3\tan \theta - \tan^3 \theta}{1 - 3\tan^2 \theta}$

Trigonometric Ratios of Half Angles

Half-angle formulas are derived from the double-angle formulas and are useful for finding the trigonometric ratios of an angle that is half of a given angle:

• $\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos \theta}{2}}$
• $\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos \theta}{2}}$
• $\tan\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} = \frac{\sin \theta}{1 + \cos \theta} = \frac{1 - \cos \theta}{\sin \theta}$

The sign depends on the quadrant in which the half-angle lies.

Table of Differences and Important Points

Function	Double Angle	Triple Angle	Half Angle
Sine	$2\sin \theta \cos \theta$	$3\sin \theta - 4\sin^3 \theta$	$\pm\sqrt{\frac{1 - \cos \theta}{2}}$
Cosine	$\cos^2 \theta - \sin^2 \theta$	$4\cos^3 \theta - 3\cos \theta$	$\pm\sqrt{\frac{1 + \cos \theta}{2}}$
Tangent	$\frac{2\tan \theta}{1 - \tan^2 \theta}$	$\frac{3\tan \theta - \tan^3 \theta}{1 - 3\tan^2 \theta}$	$\pm\sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}$

Examples

Example 1: Double Angle

Calculate $\sin(2\theta)$ and $\cos(2\theta)$ if $\sin(\theta) = \frac{1}{2}$ and $\theta$ is in the first quadrant.

Using the double angle formulas:

• $\sin(2\theta) = 2\sin \theta \cos \theta$
• $\cos(2\theta) = \cos^2 \theta - \sin^2 \theta$

Since $\sin(\theta) = \frac{1}{2}$, we can find $\cos(\theta)$ using the Pythagorean identity $\cos^2 \theta = 1 - \sin^2 \theta$:

• $\cos(\theta) = \sqrt{1 - \left(\frac{1}{2}\right)^2} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}$

Now, we can calculate:

• $\sin(2\theta) = 2 \cdot \frac{1}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}$
• $\cos(2\theta) = \left(\frac{\sqrt{3}}{2}\right)^2 - \left(\frac{1}{2}\right)^2 = \frac{3}{4} - \frac{1}{4} = \frac{1}{2}$

Example 2: Half Angle

Find $\cos\left(\frac{\theta}{2}\right)$ if $\cos(\theta) = \frac{3}{5}$ and $\frac{\theta}{2}$ is in the first quadrant.

Using the half-angle formula for cosine:

• $\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos \theta}{2}}$

Since $\frac{\theta}{2}$ is in the first quadrant, we take the positive square root:

• $\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \frac{3}{5}}{2}} = \sqrt{\frac{8}{10}} = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5}$

Conclusion

Understanding the trigonometric ratios of multiple angles is essential for solving complex problems in trigonometry. These identities allow us to express trigonometric functions of multiple angles in terms of single angles, simplifying calculations and providing a deeper insight into the properties of trigonometric functions.