Trigonometric Identity Transformations

Master the essential trigonometric identities that transform expressions involving multiple angles. These formulas are fundamental for solving complex trigonometric equations and simplifying expressions. Learn double angle formulas, half angle formulas, sum-to-product identities, and more with complete derivations and practical applications.

1. Double Angle Formulas

Expressing trigonometric functions of 2α in terms of α

Sine Double Angle

$\sin(2\alpha) = 2\sin(\alpha)\cos(\alpha)$

Derivation: From the sum formula $\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)$ , let $\beta = \alpha$ : $\sin(2\alpha) = \sin(\alpha + \alpha) = \sin(\alpha)\cos(\alpha) + \cos(\alpha)\sin(\alpha) = 2\sin(\alpha)\cos(\alpha)$

Cosine Double Angle (Three Forms)

$\cos(2\alpha) = \cos^2(\alpha) - \sin^2(\alpha)$

Basic form

$\cos(2\alpha) = 2\cos^2(\alpha) - 1$

Using $\sin^2\alpha = 1 - \cos^2\alpha$

$\cos(2\alpha) = 1 - 2\sin^2(\alpha)$

Using $\cos^2\alpha = 1 - \sin^2\alpha$

Tangent Double Angle

$\tan(2\alpha) = \frac{2\tan(\alpha)}{1 - \tan^2(\alpha)}$

Condition: $\tan(\alpha) \neq \pm 1$ (denominator must be non-zero)

Examples

\sin(2\alpha) = 2\sin(\alpha)\cos(\alpha)

\sin(60°) = 2\sin(30°)\cos(30°) = 2 \times \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}

\cos(2\alpha) = \cos^2(\alpha) - \sin^2(\alpha)

\cos(60°) = \cos^2(30°) - \sin^2(30°) = \left(\frac{\sqrt{3}}{2}\right)^2 - \left(\frac{1}{2}\right)^2 = \frac{1}{2}

\cos(2\alpha) = 2\cos^2(\alpha) - 1

\cos(60°) = 2\cos^2(30°) - 1 = 2 \times \left(\frac{\sqrt{3}}{2}\right)^2 - 1 = \frac{1}{2}

\cos(2\alpha) = 1 - 2\sin^2(\alpha)

\cos(60°) = 1 - 2\sin^2(30°) = 1 - 2 \times \left(\frac{1}{2}\right)^2 = \frac{1}{2}

2. Half Angle Formulas

Expressing trigonometric functions of α/2 in terms of α

Power Reduction Identities

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

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

These are derived by rearranging the double angle formulas for cosine.

Half Angle Formulas

$\sin(\alpha/2) = \pm\sqrt{\frac{1 - \cos \alpha}{2}}$

Sign determined by the quadrant of $\alpha/2$

$\cos(\alpha/2) = \pm\sqrt{\frac{1 + \cos \alpha}{2}}$

Sign determined by the quadrant of $\alpha/2$

$\tan(\alpha/2) = \frac{1 - \cos \alpha}{\sin \alpha} = \frac{\sin \alpha}{1 + \cos \alpha}$

Two equivalent forms that avoid the ± ambiguity

Examples

\sin(\alpha/2) = \pm\sqrt{\frac{1 - \cos \alpha}{2}}

\sin(15°) = \sqrt{\frac{1 - \cos 30°}{2}} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} = \frac{\sqrt{6} - \sqrt{2}}{4}

\cos(\alpha/2) = \pm\sqrt{\frac{1 + \cos \alpha}{2}}

\cos(15°) = \sqrt{\frac{1 + \cos 30°}{2}} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = \frac{\sqrt{6} + \sqrt{2}}{4}

\tan(\alpha/2) = \frac{1 - \cos \alpha}{\sin \alpha}

\tan(15°) = \frac{1 - \cos 30°}{\sin 30°} = \frac{1 - \frac{\sqrt{3}}{2}}{\frac{1}{2}} = 2 - \sqrt{3}

3. Sum-to-Product and Product-to-Sum Formulas

Converting between sums/differences and products of trigonometric functions

Sum-to-Product Formulas

$\sin A + \sin B = 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$

$\sin A - \sin B = 2\cos\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)$

$\cos A + \cos B = 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$

$\cos A - \cos B = -2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)$

Product-to-Sum Formulas

$\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]$

$\cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]$

$\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$

Example Applications

Sum-to-Product:

\sin 70° + \sin 10° = 2\sin(40°)\cos(30°) = 2\sin(40°) \times \frac{\sqrt{3}}{2} = \sqrt{3} \sin(40°)

Product-to-Sum:

\sin 50°\cos 20° = \frac{1}{2}[\sin(70°) + \sin(30°)] = \frac{1}{2}\left[\sin(70°) + \frac{1}{2}\right]

4. Triple Angle Formulas

Expressing trigonometric functions of 3α in terms of α

Sine Triple Angle

$\sin(3\alpha) = 3\sin(\alpha) - 4\sin^3(\alpha)$

Derivation: $\sin(3\alpha) = \sin(2\alpha + \alpha) = \sin(2\alpha)\cos(\alpha) + \cos(2\alpha)\sin(\alpha)$

Substituting formulas and simplifying leads to the cubic form.

Cosine Triple Angle

$\cos(3\alpha) = 4\cos^3(\alpha) - 3\cos(\alpha)$

Derivation: $\cos(3\alpha) = \cos(2\alpha + \alpha) = \cos(2\alpha)\cos(\alpha) - \sin(2\alpha)\sin(\alpha)$

Similar substitution and simplification process.

Example Calculation

Verify

\sin(90°)

using triple angle formula:

$\sin(90°) = \sin(3 \times 30°) = 3\sin(30°) - 4\sin^3(30°)$

$= 3 \times \frac{1}{2} - 4 \times \left(\frac{1}{2}\right)^3 = \frac{3}{2} - 4 \times \frac{1}{8} = \frac{3}{2} - \frac{1}{2} = 1$ ✓

5. Key Applications and Problem-Solving Strategies

When and how to use these identity transformations effectively

Common Problem Types

• Simplification: Reduce complex expressions using appropriate identities
• Equation solving: Transform equations to standard forms
• Integration: Power reduction for integrating trigonometric functions
• Verification: Prove trigonometric identities step by step

Strategy Selection

• Use double angle when you see 2α terms
• Use half angle when dealing with α/2 or square roots
• Use sum-to-product to simplify sums of trig functions
• Use product-to-sum for products of different angles