Reference Table of Trigonometric Values and Identities

Trigonometric Values and Identities

Introduction

This study guide summarizes key trigonometric values and identities commonly used in Precalculus. It includes exact values for sine, cosine, and tangent at notable angles, as well as important trigonometric identities and formulas. Mastery of these values and relationships is essential for solving trigonometric equations and understanding the properties of periodic functions.

Trigonometric Values for Special Angles

The following table provides the exact values of sine, cosine, and tangent for commonly used angles in radians.

Angle (Radians)	$0$	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$
$\sin \theta$	$0$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$
$\cos \theta$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$
$\tan \theta$	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	undefined

• Key Point: The values above are exact and should be memorized for efficient problem solving.

• Example: $\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$, $\tan\left(\frac{\pi}{3}\right) = \sqrt{3}$

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides are defined.

• Pythagorean Identities:

  • $\sin^2\theta + \cos^2\theta = 1$

  • $1 + \tan^2\theta = \sec^2\theta$

  • $1 + \cot^2\theta = \csc^2\theta$

• Even-Odd Identities:

  • $\sin(-\theta) = -\sin\theta$

  • $\cos(-\theta) = \cos\theta$

  • $\tan(-\theta) = -\tan\theta$

• Co-Function Identities:

  • $\sin\left(\frac{\pi}{2} - \theta\right) = \cos\theta$

  • $\cos\left(\frac{\pi}{2} - \theta\right) = \sin\theta$

  • $\tan\left(\frac{\pi}{2} - \theta\right) = \cot\theta$

Reference Table for Trigonometric Functions

The following table summarizes the values of the six trigonometric functions for the special angles $0$,$\frac{\pi}{6}$,$\frac{\pi}{4}$, $\frac{\pi}{3}$, and$\frac{\pi}{2}$.

Angle (Radians)	$0$	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$
$\sin\theta$	$0$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$
$\cos\theta$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$
$\tan\theta$	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	undefined
$\csc\theta$	undefined	$2$	$\sqrt{2}$	$\frac{2}{\sqrt{3}}$	$1$
$\sec\theta$	$1$	$\frac{2}{\sqrt{3}}$	$\sqrt{2}$	$2$	undefined
$\cot\theta$	undefined	$\sqrt{3}$	$1$	$\frac{1}{\sqrt{3}}$	$0$

• Key Point: Undefined values occur when the denominator of the function is zero (e.g., $\tan\left(\frac{\pi}{2}\right)$ is undefined because $\cos\left(\frac{\pi}{2}\right) = 0$).

• Example: $\sec\left(\frac{\pi}{3}\right) = \frac{1}{\cos\left(\frac{\pi}{3}\right)} = 2$

Applications

• These values and identities are used to solve trigonometric equations, analyze periodic phenomena, and model real-world situations involving waves and oscillations.

• Memorizing the exact values for special angles simplifies calculations in calculus, physics, and engineering.

Additional info: The original file appears to be a reference table of trigonometric values and identities, possibly extracted from a textbook or lecture notes. The content has been logically grouped and expanded for clarity and completeness.