Ratios & Identities: Trigonometric Ratios of Sub-multiple Angles

Trigonometric ratios of sub-multiple angles involve finding the sine, cosine, tangent, and other trigonometric functions of angles that are fractions of a given angle, often denoted as half-angles or third-angles. These ratios are particularly useful in solving trigonometric equations and in calculus.

Understanding Sub-multiple Angles

A sub-multiple angle is a fraction of a given angle. For example, if we have an angle $\theta$, a sub-multiple angle could be $\frac{\theta}{2}$ or $\frac{\theta}{3}$. The trigonometric ratios of these sub-multiple angles can be derived from the known trigonometric identities.

Trigonometric Identities for Sub-multiple Angles

The most commonly used sub-multiple angles are half-angles. The trigonometric identities for half-angles are derived from the double-angle formulas and the Pythagorean identity.

Half-Angle Formulas

The half-angle formulas are as follows:

1. Sine Half-Angle Formula: $$ \sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}} $$
2. Cosine Half-Angle Formula: $$ \cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}} $$
3. Tangent Half-Angle Formula: $$ \tan\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}} = \frac{\sin(\theta)}{1 + \cos(\theta)} = \frac{1 - \cos(\theta)}{\sin(\theta)} $$

The signs in the half-angle formulas depend on the quadrant in which the half-angle lies.

Table of Differences and Important Points

Trigonometric Function	Formula for Sub-multiple Angle	Important Points
Sine ($\sin$)	$\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}$	Sign depends on the quadrant of $\frac{\theta}{2}$.
Cosine ($\cos$)	$\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}$	Sign depends on the quadrant of $\frac{\theta}{2}$.
Tangent ($\tan$)	$\tan\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}$	Can be expressed in terms of sine and cosine of $\theta$.

Examples

Let's look at some examples to understand how to apply these formulas.

Example 1: Find $\sin\left(\frac{\pi}{8}\right)$

To find $\sin\left(\frac{\pi}{8}\right)$, we can use the sine half-angle formula:

$$ \sin\left(\frac{\pi}{8}\right) = \sin\left(\frac{\pi}{4}/2\right) = \pm\sqrt{\frac{1 - \cos(\pi/4)}{2}} $$

Since $\frac{\pi}{8}$ is in the first quadrant, we use the positive sign:

$$ \sin\left(\frac{\pi}{8}\right) = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{2}}{4}} = \frac{\sqrt{2 - \sqrt{2}}}{2} $$

Example 2: Find $\cos\left(\frac{\pi}{12}\right)$

To find $\cos\left(\frac{\pi}{12}\right)$, we can use the cosine half-angle formula:

$$ \cos\left(\frac{\pi}{12}\right) = \cos\left(\frac{\pi}{6}/2\right) = \pm\sqrt{\frac{1 + \cos(\pi/6)}{2}} $$

Since $\frac{\pi}{12}$ is in the first quadrant, we use the positive sign:

$$ \cos\left(\frac{\pi}{12}\right) = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{3}}{4}} = \frac{\sqrt{2 + \sqrt{3}}}{2} $$

Example 3: Verify $\tan\left(\frac{\theta}{2}\right)$ using $\theta = 60^\circ$

Let's verify the tangent half-angle formula using $\theta = 60^\circ$:

$$ \tan\left(\frac{60^\circ}{2}\right) = \tan(30^\circ) $$

Using the half-angle formula:

$$ \tan\left(\frac{60^\circ}{2}\right) = \pm\sqrt{\frac{1 - \cos(60^\circ)}{1 + \cos(60^\circ)}} = \sqrt{\frac{1 - \frac{1}{2}}{1 + \frac{1}{2}}} = \sqrt{\frac{\frac{1}{2}}{\frac{3}{2}}} = \sqrt{\frac{1}{3}} $$

Since $\tan(30^\circ) = \frac{1}{\sqrt{3}}$, the verification is correct.

Conclusion

Understanding the trigonometric ratios of sub-multiple angles is crucial for solving complex trigonometric problems. The half-angle formulas are particularly important and can be used to simplify expressions and solve equations. Remember to consider the quadrant of the angle when determining the sign of the result. With practice, these formulas become a powerful tool in the study of trigonometry.