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Values of Trigonometric Ratios for Standard Angles

Master values of trigonometric ratios for standard angles with interactive lessons and practice problems! Designed for students like you!

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Values of Trigonometric Ratios for Standard Angles

Study Guide

Key Definition

Sine, cosine, and tangent are the primary trigonometric ratios: $\sin(θ) = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos(θ) = \frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan(θ) = \frac{\text{opposite}}{\text{adjacent}}$

Important Notes

Standard angles include $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$
Radians measure angles as fractions of $\pi$ (e.g., $30^\circ = \frac{\pi}{6}$, $45^\circ = \frac{\pi}{4}$, $60^\circ = \frac{\pi}{3}$, $90^\circ = \frac{\pi}{2}$)
Learn the unit circle to understand angle measures
Key trigonometric values: $\sin(30^\circ) = \frac{1}{2}$, $\cos(60^\circ) = \frac{1}{2}$
Use mnemonic 'SOH-CAH-TOA' for remembering ratios

Mathematical Notation

$\sin(\theta)$ is the sine function

$\cos(\theta)$ is the cosine function

$\tan(\theta)$ is the tangent function

$\frac{a}{b}$ represents a fraction

$\sqrt{x}$ represents the square root of $x$

Use parentheses in function notation (e.g., sin(θ)) and specify units (° or rad).

Why It Works

Understanding the relationships between angles and sides in right triangles allows for calculating unknown lengths using $\sin$, $\cos$, and $\tan$.

Remember

Trigonometric values for standard angles like $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, and $90^\circ$ are fundamental.

Quick Reference

Sine of 30°:$\sin(30^\circ) = \frac{1}{2}$

Cosine of 60°:$\cos(60^\circ) = \frac{1}{2}$

Tangent of 45°:$\tan(45^\circ) = 1$

Sine of 0° / 0:$\sin(0^\circ) = 0, \quad \sin(0) = 0$

Cosine of 0° / 0:$\cos(0^\circ) = 1, \quad \cos(0) = 1$

Sine of 90° / π/2:$\sin(90^\circ) = 1, \quad \sin\bigl(\tfrac{\pi}{2}\bigr) = 1$

Cosine of 90° / π/2:$\cos(90^\circ) = 0, \quad \cos\bigl(\tfrac{\pi}{2}\bigr) = 0$

Understanding Values of Trigonometric Ratios for Standard Angles

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Video explanation of this concept

Beginner

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Beginner Explanation

In a right triangle relative to an angle θ, the sine is the ratio of the opposite side to the hypotenuse. For example, $\sin(θ) = \frac{\text{opposite}}{\text{hypotenuse}}$. If the hypotenuse is 2 and the opposite side is 1, then $\sin(30^\circ) = \frac{1}{2}$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is $\sin(30^\circ)$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Calculate the height of a tree using $\tan(\theta)$ with a measured distance of 10 meters and an angle of elevation $45^\circ$.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Determine $\cos(60^\circ)$ using a right triangle with hypotenuse 2 and adjacent side 1 (a 30°–60°–90° triangle).

Challenge Quiz

Single Choice Quiz

Advanced

What is the value of $\tan(30^\circ)$?

Recap

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