Analytic Trigonometry: Double-Angle, Power-Reducing, and Half-Angle Formulas

Analytic Trigonometry

Double-Angle, Power-Reducing, and Half-Angle Formulas

This section explores advanced trigonometric identities used to simplify expressions and solve equations. The focus is on double-angle, power-reducing, and half-angle formulas, which are essential tools in precalculus for manipulating trigonometric functions.

Double-Angle Formulas

Double-angle formulas express trigonometric functions of twice an angle in terms of functions of the original angle. These are useful for finding exact values and simplifying expressions.

• Sine Double-Angle Formula: $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$

• Cosine Double-Angle Formulas:

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

  • $\cos(2\theta) = 2\cos^2(\theta) - 1$

  • $\cos(2\theta) = 1 - 2\sin^2(\theta)$

• Tangent Double-Angle Formula: $\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$

Example: If $\theta$ lies in quadrant II and $\sin(\theta) = \frac{3}{5}$, find $\sin(2\theta)$ using the double-angle formula.

Verifying Trigonometric Identities

Trigonometric identities can be verified by algebraic manipulation, substitution, and using known formulas. Double-angle formulas are often used in these verifications.

• Key Point: To verify an identity, rewrite one or both sides using double-angle formulas and simplify.

• Example: Verify $\cos(2\theta) = 2\cos^2(\theta) - 1$ by expanding $\cos^2(\theta)$ and $\sin^2(\theta)$.

Power-Reducing Formulas

Power-reducing formulas express powers of sine and cosine in terms of the first power, making them useful for integration and simplification.

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

• $\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$

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

Example: Write $\sin^2(\theta)$ as an equivalent expression without powers greater than 1 using the power-reducing formula.

Half-Angle Formulas

Half-angle formulas allow calculation of trigonometric functions of half an angle. The sign of the result depends on the quadrant in which the half-angle lies.

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

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

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

Key Point: The $\pm$ symbol does not mean there are two possible values; it indicates the sign must be determined based on the quadrant.

Example: Use the half-angle formula to find the exact value of $\sin\left(\frac{\theta}{2}\right)$ given $\cos(\theta)$ and the quadrant.

Half-Angle Formulas for Tangent

Half-angle formulas for tangent are derived from the sine and cosine half-angle formulas:

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

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