Trigonometry - Compound Angle Formulas and Ratios

Vocabulary flashcards covering trigonometric identities, mnemonic definitions, quadrant rules, and standard angle properties from Example 8 Exercise 1-2.

Sine Addition Formula

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

Tangent Addition Formula

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

Pythagoras Theorem

$$c^2 = a^2 + b^2$$

SOH

Trigonometric ratio mnemonic defining $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$

CAH

Trigonometric ratio mnemonic defining $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

TOA

Trigonometric ratio mnemonic defining $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

CAST Rule

A four-quadrant rule indicating where trigonometric functions are positive: Quadrant $I$ (All), Quadrant $II$ (Sine), Quadrant $III$ (Tangent), Quadrant $IV$ (Cosine).

Standard Position Angle $A$

Given $\sin(A) = \frac{4}{5}$ in quadrant $II$ ($90^\circ < A < 180^\circ$), the remaining ratios are $\cos(A) = -\frac{3}{5}$ and $\tan(A) = -\frac{4}{3}$.

Standard Position Angle $B$

Given $\cos(B) = -\frac{5}{13}$ in quadrant $III$ ($180^\circ < B < 270^\circ$), the remaining ratios are $\sin(B) = -\frac{12}{13}$ and $\tan(B) = \frac{12}{5}$.

Calculated Value of $\sin(A+B)$

$$\sin(A+B) = \frac{16}{65}$$