Unit Circle and Trigonometry Review

Vocabulary flashcards covering the unit circle, special right triangles, trigonometric functions, reciprocal identities, double angle formulas, and inverse trigonometric function restrictions.

All students take calculus

A mnemonic for remembering which trigonometric functions are positive in each quadrant: Quadrant 1 (All functions), Quadrant 2 (Sine), Quadrant 3 (Tangent), and Quadrant 4 (Cosine).

Unit circle coordinate associations

The convention where the $x$-coordinate corresponds to $\cos(\theta)$ and the $y$-coordinate corresponds to $\sin(\theta)$, giving points on the unit circle as $(x, y) = (\cos(\theta), \sin(\theta))$.

Reference angle

The acute angle formed between the hypotenuse (terminal side) of an angle and the $x$-axis, always having a positive measure between $0^\circ$ and $90^\circ$.

Reference angle formulas by quadrant

Formulas to find the reference angle from a given angle: in Quadrant 1 it is the angle itself; in Quadrant 2 it is $180^\circ - \text{angle}$; in Quadrant 3 it is $\text{angle} - 180^\circ$; in Quadrant 4 it is $360^\circ - \text{angle}$.

Coterminal angles

Angles that land in the exact same position on the unit circle and differ by complete rotations of $360^\circ$ (or $2\pi$ radians).

Direction of angle rotation

The convention where a positive angle rotates counterclockwise (opposite the direction of a clock) and a negative angle rotates clockwise (following the direction of a clock).

Radians to degrees conversion

The method of converting an angle given in radians to degrees by multiplying the radian measure by $\frac{180^\circ}{\pi}$.

30-60-90 triangle

A special right triangle whose side opposite the $30^\circ$ angle is $1$, side opposite the $60^\circ$ angle is $\sqrt{3}$, and hypotenuse opposite the $90^\circ$ angle is $2$.

45-45-90 triangle

A special right triangle whose sides opposite each $45^\circ$ angle are $1$, and whose hypotenuse opposite the $90^\circ$ angle is $\sqrt{2}$.

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 $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$.

Secant ($\sec(\theta)$)

The reciprocal trigonometric function of cosine, defined as $\sec(\theta) = \frac{1}{\cos(\theta)}$ or $\frac{\text{hypotenuse}}{\text{adjacent}}$.

Cosecant ($\csc(\theta)$)

The reciprocal trigonometric function of sine, defined as $\csc(\theta) = \frac{1}{\sin(\theta)}$ or $\frac{\text{hypotenuse}}{\text{opposite}}$.

Cotangent ($\cot(\theta)$)

The reciprocal trigonometric function of tangent, defined as $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}$ or $\frac{\text{adjacent}}{\text{opposite}}$.

Undefined quadrantal values

Trigonometric ratios resulting in division by zero on the coordinate axes: $\tan(90^\circ)$ and $\tan(270^\circ)$ (where $x = 0$), as well as $\cot(0^\circ)$ and $\cot(180^\circ)$ (where $y = 0$).

Pythagorean triples (special triangles)

Sets of three integer side lengths for right triangles referenced in the lecture: $3-4-5$, $5-12-13$, $8-15-17$, $7-24-25$, $9-40-41$, and $11-60-61$.

Double angle formula for sine

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

Double angle formula for cosine

The trigonometric identity expressing the cosine of twice an angle as $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$.

Inverse sine restricted range

The restricted output range of the inverse sine function, limited to Quadrants 1 and 4 between $-90^\circ$ and $90^\circ$.

Inverse cosine restricted range

The restricted output range of the inverse cosine function, limited to Quadrants 1 and 2 between $0^\circ$ and $180^\circ$.

Inverse tangent restricted range

The restricted output range of the inverse tangent function, limited to Quadrants 1 and 4 between $-90^\circ$ and $90^\circ$.