Trigonometry - Unit 10 Formula Sheet

Comprehensive vocabulary flashcards covering fundamental trigonometric definitions, identities, quadrant rules, angle formulas, and transformation laws from the Unit 10 reference sheet.

SOH — CAH — TOA

A mnemonic defining the primary right-triangle trigonometric ratios: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$, $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$, and $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$.

Reciprocal Identities

Identities formed by inverting ratios: $\csc(\theta) = \frac{1}{\sin(\theta)}$, $\sec(\theta) = \frac{1}{\cos(\theta)}$, and $\cot(\theta) = \frac{1}{\tan(\theta)}$, satisfying $\sin(\theta)\csc(\theta) = 1$, $\cos(\theta)\sec(\theta) = 1$, and $\tan(\theta)\cot(\theta) = 1$.

Quotient Identities

Identities expressing tangent and cotangent in terms of sine and cosine: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ and $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$.

Pythagorean Identities

Three fundamental trigonometric relationships: $\sin^2(\theta) + \cos^2(\theta) = 1$, $1 + \tan^2(\theta) = \sec^2(\theta)$, and $1 + \cot^2(\theta) = \csc^2(\theta)$.

Fundamental Law of Trigonometry

The subtraction formula for cosine, $\cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)$, from which the other compound-angle identities are derived.

Compound-Angle Identities for Sine

Addition and subtraction formulas for sine: $\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)$ and $\sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta)$.

Compound-Angle Identities for Cosine

Addition and subtraction formulas for cosine: $\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)$ and $\cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)$.

Compound-Angle Identities for Tangent

Addition and subtraction formulas for tangent: $\tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha)\tan(\beta)}$ and $\tan(\alpha - \beta) = \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)}$.

Allied Angles Rule

When $\theta$ is added to or subtracted from an odd multiple of $90^\circ$, the function changes to its co-ratio ($\sin \leftrightarrow \cos$, $\tan \leftrightarrow \cot$, $\sec \leftrightarrow \csc$); if an even multiple of $90^\circ$ is involved, the ratio type remains unchanged. The resulting sign is determined by the quadrant of the original angle.

ASTC Rule ("Add Sugar To Coffee")

A memory cue indicating which ratios are positive by quadrant: Quadrant I ($0^\circ$ to $90^\circ$): All positive; Quadrant II ($90^\circ$ to $180^\circ$): $\sin(\theta)$ and $\csc(\theta)$ positive; Quadrant III ($180^\circ$ to $270^\circ$): $\tan(\theta)$ and $\cot(\theta)$ positive; Quadrant IV ($270^\circ$ to $360^\circ$): $\cos(\theta)$ and $\sec(\theta)$ positive.

Fast Construction Pattern for Standard Angles

The technique for constructing $\sin(\theta)$ at $0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$ using $\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}$, taking the sequence in reverse for $\cos(\theta)$, and using quotients or reciprocals for the remaining functions.

Double-Angle Identity for Sine

The identity expressing the sine of twice an angle: $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$.

Double-Angle Identities for Cosine

The three interchangeable forms: $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$, $\cos(2\theta) = 2\cos^2(\theta) - 1$, and $\cos(2\theta) = 1 - 2\sin^2(\theta)$.

Double-Angle Identity for Tangent

The identity expressing the tangent of twice an angle: $\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$.

Half-Angle Identities for Sine and Cosine

Formulas for half-angles: $\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}$ and $\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}$, with the sign chosen based on the quadrant of $\frac{\theta}{2}$.

Half-Angle Identities for Tangent

The radical and quotient expressions for the tangent of a half-angle: $\tan\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}$, which also equals $\frac{\sin(\theta)}{1 + \cos(\theta)}$ and $\frac{1 - \cos(\theta)}{\sin(\theta)}$.

Triple-Angle Identities

Identities for multiples of three: $\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)$, $\cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta)$, and $\tan(3\theta) = \frac{3\tan(\theta) - \tan^3(\theta)}{1 - 3\tan^2(\theta)}$.

Product-to-Sum Identities

Formulas converting products of sines and cosines into sums or differences: $2\sin(\alpha)\cos(\beta) = \sin(\alpha + \beta) + \sin(\alpha - \beta)$, $2\cos(\alpha)\sin(\beta) = \sin(\alpha + \beta) - \sin(\alpha - \beta)$, $2\cos(\alpha)\cos(\beta) = \cos(\alpha + \beta) + \cos(\alpha - \beta)$, and $2\sin(\alpha)\sin(\beta) = \cos(\alpha - \beta) - \cos(\alpha + \beta)$.

Sum-to-Product Identities

Formulas converting sums and differences of sines and cosines into products: $\sin(P) + \sin(Q) = 2\sin\left(\frac{P + Q}{2}\right)\cos\left(\frac{P - Q}{2}\right)$, $\sin(P) - \sin(Q) = 2\cos\left(\frac{P + Q}{2}\right)\sin\left(\frac{P - Q}{2}\right)$, $\cos(P) + \cos(Q) = 2\cos\left(\frac{P + Q}{2}\right)\cos\left(\frac{P - Q}{2}\right)$, and $\cos(P) - \cos(Q) = -2\sin\left(\frac{P + Q}{2}\right)\sin\left(\frac{P - Q}{2}\right)$.

Distance Formula Link

The formula $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ for points $P(x_1, y_1)$ and $Q(x_2, y_2)$, utilized with coordinates containing sine and cosine to establish the fundamental trigonometric law.

One-Line Memory Map

The procedural sequence for navigating trigonometry topics: Ratios $\rightarrow$ Pythagorean $\rightarrow$ Compound angles $\rightarrow$ Allied angles $\rightarrow$ Double $\rightarrow$ Half $\rightarrow$ Triple $\rightarrow$ Product/Sum transformations.