Trigonometry Identities, Values, and Rules

Vocabulary flashcards reviewing core trigonometric identities, exact values, triangle rules, graph properties, and equation solving techniques.

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

The fundamental trigonometric identity equal to $1$.

$\sin^2(\theta)$ (Meaning)

$(\sin(\theta))^2$, which denotes the sine of $\theta$, squared; it does not mean $\sin(\theta^2)$.

$\sin^2(\theta)$ (Subject Form)

$\sin^2(\theta) = 1 - \cos^2(\theta)$.

$\cos^2(\theta)$ (Subject Form)

$\cos^2(\theta) = 1 - \sin^2(\theta)$.

Quotient Identity for $\tan(\theta)$

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

SOH

Trigonometric mnemonic standing for $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$.

CAH

Trigonometric mnemonic standing for $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$.

TOA

Trigonometric mnemonic standing for $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$.

Exact Values of $\sin(\theta)$

$\sin(0^\circ) = 0$, $\sin(30^\circ) = \frac{1}{2}$, $\sin(45^\circ) = \frac{\sqrt{2}}{2}$ (or $\frac{1}{\sqrt{2}}$), $\sin(60^\circ) = \frac{\sqrt{3}}{2}$, and $\sin(90^\circ) = 1$.

Exact Values of $\cos(\theta)$

$\cos(0^\circ) = 1$, $\cos(30^\circ) = \frac{\sqrt{3}}{2}$, $\cos(45^\circ) = \frac{\sqrt{2}}{2}$ (or $\frac{1}{\sqrt{2}}$), $\cos(60^\circ) = \frac{1}{2}$, and $\cos(90^\circ) = 0$.

Exact Values of $\tan(\theta)$

$\tan(0^\circ) = 0$, $\tan(30^\circ) = \frac{1}{\sqrt{3}}$ (or $\frac{\sqrt{3}}{3}$), $\tan(45^\circ) = 1$, $\tan(60^\circ) = \sqrt{3}$, and $\tan(90^\circ) = \text{Undefined}$.

The Sine Rule (Finding a Side)

$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$.

The Sine Rule (Finding an Angle)

$\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}$.

Conditions for Using the Sine Rule

Used when you know a side and its opposite angle, plus one more side or angle.

Ambiguous Case of the Sine Rule

When finding an angle, $\sin(\theta) = k$ gives two possible angles: $\theta$ and $180^\circ - \theta$; you must check whether both fit in the triangle.

The Cosine Rule (Finding a Side)

$a^2 = b^2 + c^2 - 2bc\cos(A)$.

The Cosine Rule (Finding an Angle)

$\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}$.

Conditions for Using the Cosine Rule

Used when you know two sides and the angle between them (SAS), or all three sides (SSS).

Area of a Triangle Using Sine

$\text{Area} = \frac{1}{2}ab\sin(C)$, where $C$ is the angle between sides $a$ and $b$.

Supplementary Angle Identities

$\sin(180^\circ - \theta) = \sin(\theta)$, $\cos(180^\circ - \theta) = -\cos(\theta)$, and $\tan(180^\circ - \theta) = -\tan(\theta)$.

Negative Angle Identities

$\sin(-\theta) = -\sin(\theta)$, $\cos(-\theta) = \cos(\theta)$, and $\tan(-\theta) = -\tan(\theta)$.

Complementary Angle Identities

$\sin(90^\circ - \theta) = \cos(\theta)$ and $\cos(90^\circ - \theta) = \sin(\theta)$.

Period of $y = \sin(x)$ and $y = \cos(x)$

$360^\circ$, which implies $\sin(\theta + 360^\circ) = \sin(\theta)$ and $\cos(\theta + 360^\circ) = \cos(\theta)$.

Period of $y = \tan(x)$

$180^\circ$, which implies $\tan(\theta + 180^\circ) = \tan(\theta)$.

Range of $y = \sin(x)$ and $y = \cos(x)$

$-1 \le y \le 1$.

Range of $y = \tan(x)$

All real numbers.

Asymptotes of $y = \tan(x)$ ($0^\circ$ to $360^\circ$)

$x = 90^\circ$ and $x = 270^\circ$.

$y$-Intercept of $y = \sin(x)$

At the origin $(0, 0)$.

$y$-Intercept of $y = \cos(x)$

At $(0, 1)$.

CAST Diagram

A reference showing which trig functions are positive in each quadrant: All ($0^\circ\text{–}90^\circ$), Sin ($90^\circ\text{–}180^\circ$), Tan ($180^\circ\text{–}270^\circ$), and Cos ($270^\circ\text{–}360^\circ$).

Second Solution for $\sin(\theta) = k$

$\theta_2 = 180^\circ - \theta_1$, then add or subtract $360^\circ$ for more solutions within the range.

Second Solution for $\cos(\theta) = k$

$\theta_2 = 360^\circ - \theta_1$, then add or subtract $360^\circ$ for more solutions within the range.

Solutions for $\tan(\theta) = k$

Add or subtract $180^\circ$ to the first solution.

Strategy for Mixed Trigonometric Equations

Use $\sin^2(\theta) + \cos^2(\theta) = 1$ or $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ to rewrite equations mixing $\sin^2$, $\cos^2$, and $\tan$ in terms of one trig function.

Initial Step for Solving $\sin(\theta) = k$

Use the inverse function ($\sin^{-1}(k)$) to find the principal value, then use the graph or CAST diagram for the other solutions.

Common Mistake When Solving Trig Equations

Forgetting the second solution, or giving answers outside the stated range.