Evaluating Trigonometric Functions for Special Angles

Evaluating Trigonometric Functions for Special Angles

Introduction

Understanding how to evaluate trigonometric functions at special angles is a foundational skill in trigonometry. These angles, often called "reference angles," include $0$,$\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$, and their multiples. Mastery of these values is essential for solving equations, graphing functions, and applying trigonometry in various contexts.

Key Concepts

• Trigonometric Functions: The six basic trigonometric functions are sine ($\sin$), cosine ($\cos$), tangent ($\tan$), cosecant ($\csc$), secant ($\sec$), and cotangent ($\cot$).

• Special Angles: Common angles used in trigonometry include $0$,$\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$, and their corresponding degree measures ($0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$).

• Unit Circle: The unit circle provides the values of trigonometric functions for these special angles.

Evaluating Trigonometric Functions

To evaluate trigonometric functions at special angles, use the unit circle or memorized values. Below are the standard values for sine and cosine at the most common special angles:

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

Examples

• Example 1: $\sin\left(\frac{\pi}{2}\right)$ Solution: $\sin\left(\frac{\pi}{2}\right) = 1$

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

• Example 3: $\tan\left(\frac{\pi}{4}\right)$ Solution: $\tan\left(\frac{\pi}{4}\right) = 1$

• Example 4: $\sin\left(\frac{\pi}{6}\right)$ Solution: $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$

Applications

• Solving trigonometric equations

• Graphing trigonometric functions

• Modeling periodic phenomena in physics and engineering

Additional info: The problems in the image ask for the evaluation of trigonometric functions at special angles, which is a core skill in trigonometry and directly related to the study of trigonometric functions and their properties.