Double-Angle and Half-Angle Formulas in Analytic Trigonometry

Double-Angle and Half-Angle Formulas

Double-Angle Formulas

The double-angle formulas are fundamental identities in trigonometry that express trigonometric functions of twice an angle in terms of functions of the original angle. These formulas are useful for simplifying expressions and solving equations involving trigonometric functions.

• 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: Find the exact value of $\sin(2\theta)$ and $\cos(2\theta)$ given $\sin\theta = \frac{3}{5}$ and $0 < \theta < \frac{\pi}{2}$.

• $\cos\theta = \sqrt{1 - \sin^2\theta} = \sqrt{1 - (\frac{3}{5})^2} = \frac{4}{5}$

• $\sin(2\theta) = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} = \frac{24}{25}$

• $\cos(2\theta) = 2 \cdot (\frac{4}{5})^2 - 1 = 2 \cdot \frac{16}{25} - 1 = \frac{32}{25} - 1 = \frac{7}{25}$

Applications of Double-Angle Formulas

Double-angle formulas are used to simplify trigonometric expressions, solve equations, and analyze periodic phenomena. They are also essential in calculus and physics for integrating and differentiating trigonometric functions.

• Solving Equations: For example, to solve $\sin(2\theta) = 0$, set $2\theta = n\pi$ and solve for $\theta$.

• Expressing Powers: $\sin^2\theta$ and $\cos^2\theta$ can be rewritten using double-angle formulas:

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

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

Half-Angle Formulas

Half-Angle Formulas for Sine, Cosine, and Tangent

Half-angle formulas allow us to express trigonometric functions of half an angle in terms of the original angle. These are particularly useful for finding exact values of trigonometric functions at specific angles and for simplifying integrals.

• Sine Half-Angle Formula: $\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos\theta}{2}}$

• Cosine Half-Angle Formula: $\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos\theta}{2}}$

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

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

Example: Find the exact value of $\cos(15^\circ)$ using the half-angle formula.

• $\cos(15^\circ) = \cos\left(\frac{30^\circ}{2}\right) = \sqrt{\frac{1 + \cos(30^\circ)}{2}}$

• $\cos(30^\circ) = \frac{\sqrt{3}}{2}$

• $\cos(15^\circ) = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{3}}{4}} = \frac{\sqrt{2 + \sqrt{3}}}{2}$

Choosing the Correct Sign in Half-Angle Formulas

The sign (positive or negative) in the half-angle formulas depends on the quadrant in which the resulting angle lies. For example, $\sin\left(\frac{\theta}{2}\right)$ is positive if $0 < \frac{\theta}{2} < \pi$, and negative otherwise.

• Quadrant Analysis:

  • First and second quadrants: sine is positive

  • First and fourth quadrants: cosine is positive

  • Third and fourth quadrants: sine is negative

  • Second and third quadrants: cosine is negative

Half-Angle Formulas for Tangent

The tangent half-angle formula can be written in several equivalent forms, which are useful for solving equations and simplifying expressions.

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

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

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

Example: If $\cos\alpha = \frac{3}{5}$ and $0 < \alpha < \frac{\pi}{2}$, find the exact value of $\sin\left(\frac{\alpha}{2}\right)$, $\cos\left(\frac{\alpha}{2}\right)$, and $\tan\left(\frac{\alpha}{2}\right)$.

• $\sin\left(\frac{\alpha}{2}\right) = \sqrt{\frac{1 - \frac{3}{5}}{2}} = \sqrt{\frac{2}{10}} = \frac{1}{\sqrt{5}}$

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

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

Summary Table: Double-Angle and Half-Angle Formulas

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

Additional info: The notes also include quadrant analysis for sign determination and step-by-step worked examples for both double-angle and half-angle formulas, reinforcing their application in solving trigonometric equations and finding exact values.