Trigonometric Identities

A complete set of vocabulary flashcards covering basic, double-angle, power-reducing, half-angle, product-to-sum, sum-to-product, and triple-angle trigonometric identities.

Sine Sum Identity $\sin(A + B)$

$$\sin(A)\cos(B) + \cos(A)\sin(B)$$

Sine Difference Identity $\sin(A - B)$

$$\sin(A)\cos(B) - \cos(A)\sin(B)$$

Cosine Sum Identity $\cos(A + B)$

$$\cos(A)\cos(B) - \sin(A)\sin(B)$$

Cosine Difference Identity $\cos(A - B)$

$$\cos(A)\cos(B) + \sin(A)\sin(B)$$

Tangent Sum Identity $\tan(A + B)$

$$\frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)}$$

Tangent Difference Identity $\tan(A - B)$

$$\frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)}$$

Sine Double-Angle Identity $\sin(2x)$

$$2\sin(x)\cos(x)$$

Cosine Double-Angle Identity (in terms of sine and cosine)

$$\cos^2(x) - \sin^2(x)$$

Cosine Double-Angle Identity (strictly in terms of cosine)

$$2\cos^2(x) - 1$$

Cosine Double-Angle Identity (strictly in terms of sine)

$$1 - 2\sin^2(x)$$

Tangent Double-Angle Identity $\tan(2x)$

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

Power-Reducing Formula for $\sin^2(x)$

$$\frac{1 - \cos(2x)}{2}$$

Power-Reducing Formula for $\cos^2(x)$

$$\frac{1 + \cos(2x)}{2}$$

Power-Reducing Formula for $\tan^2(x)$

$$\frac{1 - \cos(2x)}{1 + \cos(2x)}$$

Half-Angle Sign Determination

The quadrant where the half-angle $\frac{x}{2}$ lies, NOT where $x$ lies.

Sine Half-Angle Formula $\sin\left(\frac{x}{2}\right)$

$$\pm \sqrt{\frac{1 - \cos(x)}{2}}$$

Cosine Half-Angle Formula $\cos\left(\frac{x}{2}\right)$

$$\pm \sqrt{\frac{1 + \cos(x)}{2}}$$

Radical Tangent Half-Angle Formula $\tan\left(\frac{x}{2}\right)$

$$\pm \sqrt{\frac{1 - \cos(x)}{1 + \cos(x)}}$$

Non-Radical Tangent Half-Angle Formulas $\tan\left(\frac{x}{2}\right)$

$\frac{1 - \cos(x)}{\sin(x)}$ and $\frac{\sin(x)}{1 + \cos(x)}$

Product-to-Sum Formula for $\sin(A)\cos(B)$

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

Product-to-Sum Formula for $\cos(A)\sin(B)$

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

Product-to-Sum Formula for $\cos(A)\cos(B)$

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

Product-to-Sum Formula for $\sin(A)\sin(B)$

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

Sum-to-Product Formula for $\sin(x) + \sin(y)$

$$2\sin\left(\frac{x + y}{2}\right)\cos\left(\frac{x - y}{2}\right)$$

Sum-to-Product Formula for $\sin(x) - \sin(y)$

$$2\cos\left(\frac{x + y}{2}\right)\sin\left(\frac{x - y}{2}\right)$$

Sum-to-Product Formula for $\cos(x) + \cos(y)$

$$2\cos\left(\frac{x + y}{2}\right)\cos\left(\frac{x - y}{2}\right)$$

Sum-to-Product Formula for $\cos(x) - \cos(y)$

$$-2\sin\left(\frac{x + y}{2}\right)\sin\left(\frac{x - y}{2}\right)$$

Cosine Triple-Angle Expansion $\cos(3x)$

$$4\cos^3(x) - 3\cos(x)$$

Sine Triple-Angle Expansion $\sin(3x)$

$$3\sin(x) - 4\sin^3(x)$$