Essential Trigonometric Values and Identities

Trigonometric Functions: Key Angles and Values

Common Angles in Degrees and Radians

Understanding the values of trigonometric functions at standard angles is fundamental in precalculus and trigonometry. The following table summarizes the sine, cosine, and tangent values for commonly used angles, along with their degree and radian measures.

Angle (Degrees)	Angle (Radians)	sin(θ)	cos(θ)	tan(θ)
0°	$0$	$0$	$1$	$0$
30°	$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$
45°	$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
60°	$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
90°	$\frac{\pi}{2}$	$1$	$0$	undefined
120°	$\frac{2\pi}{3}$	$\frac{\sqrt{3}}{2}$	$-\frac{1}{2}$	$-\sqrt{3}$
135°	$\frac{3\pi}{4}$	$\frac{\sqrt{2}}{2}$	$-\frac{\sqrt{2}}{2}$	$-1$
150°	$\frac{5\pi}{6}$	$\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$	$-\frac{\sqrt{3}}{3}$
180°	$\pi$	$0$	$-1$	$0$
270°	$\frac{3\pi}{2}$	$-1$	$0$	undefined
360°	$2\pi$	$0$	$1$	$0$

Reciprocal Trigonometric Identities

Each basic trigonometric function has a reciprocal function. These identities are essential for simplifying expressions and solving equations.

• Cosecant: $\csc\theta = \frac{1}{\sin\theta}$

• Secant: $\sec\theta = \frac{1}{\cos\theta}$

• Cotangent: $\cot\theta = \frac{1}{\tan\theta}$

Memory Patterns for Sine and Cosine

To quickly recall the values of sine and cosine for the key angles in the first quadrant, observe the following patterns:

• Sine values (0° to 90°): $0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1$

• Cosine values (0° to 90°): $1, \frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2}, \frac{1}{2}, 0$

These patterns help in memorizing the trigonometric values efficiently.

Example: Evaluating Trigonometric Functions

• Example 1: Find $\sin 45^\circ$ and $\cos 45^\circ$.

  • From the table: $\sin 45^\circ = \frac{\sqrt{2}}{2}$, $\cos 45^\circ = \frac{\sqrt{2}}{2}$

• Example 2: Find $\tan 60^\circ$.

  • From the table: $\tan 60^\circ = \sqrt{3}$

• Example 3: Find $\sec 30^\circ$.

  • First, $\cos 30^\circ = \frac{\sqrt{3}}{2}$

  • So, $\sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$