Mathematics In Physics: Trigonometry Review

These vocabulary flashcards cover fundamental trigonometric definitions, ratios, quadrant rules, identities, and transformation formulas based on the physics mathematics lecture notes.

Angle ($\theta$)

The relationship defined by the formula $\theta = \frac{S}{r}$ where $S$ is the arc and $r$ is the radius; this is true for radian measurement only.

Radian and Degree Relation

The mathematical conversion where $2\pi\,\text{radian} = 360^{\circ}$ and $1\,radian = 57.3^{\circ}$.

Hypotenuse

In a right-angled triangle, it is the largest side opposite to the right angle.

Perpendicular

The side of a right-angled triangle that is opposite to the angle $\theta$ being considered.

Base

The side of a right-angled triangle adjacent to the angle $\theta$ other than the hypotenuse.

sin $\theta$

The trigonometric ratio defined as $\frac{\text{Perpendicular}}{\text{Hypotenuse}}$ or $\frac{AB}{AC}$.

cos $\theta$

The trigonometric ratio defined as $\frac{\text{Base}}{\text{Hypotenuse}}$ or $\frac{BC}{AC}$.

tan $\theta$

The trigonometric ratio defined as $\frac{\text{Perpendicular}}{\text{Base}}$ or $\frac{AB}{BC}$.

cosec $\theta$

The reciprocal of sine, defined as $\frac{\text{Hypotenuse}}{\text{Perpendicular}}$ or $\frac{AC}{AB}$.

sec $\theta$

The reciprocal of cosine, defined as $\frac{\text{Hypotenuse}}{\text{Base}}$ or $\frac{AC}{BC}$.

cot $\theta$

The reciprocal of tangent, defined as $\frac{\text{Base}}{\text{Perpendicular}}$ or $\frac{BC}{AB}$.

Range of Sine and Cosine

The values for these ratios always lie between $-1$ and $+1$.

Range of Secant and Cosecant

The numerical values for these ratios cannot be less than one.

First Quadrant

The quadrant where all trigonometric ratios are positive; includes angles such as $(90^{\circ} - \theta)$.

Second Quadrant

The quadrant where only sine and cosec are positive; includes angles such as $(90^{\circ} + \theta)$ and $(180^{\circ} - \theta)$.

Third Quadrant

The quadrant where only tan and cot are positive; includes angles such as $(180^{\circ} + \theta)$ and $(270^{\circ} - \theta)$.

Fourth Quadrant

The quadrant where only cos and sec are positive; includes angles such as $(270^{\circ} + \theta)$ and $(0^{\circ} - \theta)$.

Fundamental Pythagorean Identity

The trigonometric relation stating that $\sin^{2}(\theta) + \cos^{2}(\theta) = 1$.

Parent Angles 90° or 270°

Angles that cause sin to change to cos, tan to change to cot, and sec to change to cosec.

Parent Angles 180° or 360°

Angles that result in no change to the trigonometric function when calculating allied angles.

sin(A + B)

The addition formula defined as $\sin(A)\cos(B) + \cos(A)\sin(B)$.

cos(A + B)

The addition formula defined as $\cos(A)\cos(B) - \sin(A)\sin(B)$.

sin 2A

A double angle formula defined as $2\sin(A)\cos(A)$.

cos 2A

A double angle formula defined as $\cos^{2}(A) - \sin^{2}(A)$ or $1 - 2\sin^{2}(A)$ or $2\cos^{2}(A) - 1$.