Trigonometric Ratios and Identities: Lecture 2

Vocabulary and formula flashcards covering fundamental trigonometric identities, quotient and reciprocal relations, negative angle rules, quadrant sign conventions, and triangle area.

First Fundamental Identity

$$\sin^2(\theta) + \cos^2(\theta) = 1$$

Second Fundamental Identity

$$1 + \tan^2(\theta) = \sec^2(\theta)$$

Third Fundamental Identity

$$1 + \cot^2(\theta) = \csc^2(\theta)$$

Quotient Identities

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ and $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$

Reciprocal Identity for $\cot(\theta)$

$$\cot(\theta) = \frac{1}{\tan(\theta)}$$

Triangle Area Formula (Given Two Sides and Included Angle)

$$\text{Area} = \frac{1}{2}ab\sin(\theta)$$

Negative Angle Identities for Cosine and Secant

$\cos(-\theta) = \cos(\theta)$ and $\sec(-\theta) = \sec(\theta)$

Negative Angle Identities for Sine, Cosecant, Tangent, and Cotangent

$\sin(-\theta) = -\sin(\theta)$, $\csc(-\theta) = -\csc(\theta)$, $\tan(-\theta) = -\tan(\theta)$, and $\cot(-\theta) = -\cot(\theta)$

Quadrant I Sign Convention

All trigonometric functions are positive ($+ve$).

Quadrant II Sign Convention

Only $\sin$ and $\csc$ are positive ($+ve$).

Quadrant III Sign Convention

Only $\tan$ and $\cot$ are positive ($+ve$).

Quadrant IV Sign Convention

Only $\cos$ and $\sec$ are positive ($+ve$).