Values of Trigonometric Ratios for Common Angles

Study Guide

Key Definition

The sine of an angle is the trigonometric ratio of the opposite side to the hypotenuse: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$. The cosine of an angle is the trigonometric ratio of the adjacent side to the hypotenuse: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$. The tangent of an angle is the trigonometric ratio of the opposite side to the adjacent side: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$.

Important Notes

The sine of $0^\circ$ is $0$
The cosine of $90^\circ$ is $0$
The tangent of $45^\circ$ is $1$
Values of trigonometric functions repeat every $360^\circ$
Undefined values occur when dividing by zero, as in $\tan(90^\circ)$

Mathematical Notation

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

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

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

$\pi$ equals approximately 3.14159

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

Remember to use proper notation when solving problems

Why It Works

Trigonometric ratios can be understood both from right-triangle geometry and the unit circle. In the unit circle, each angle $\theta$ maps to a point $(\cos\theta,\,\sin\theta)$. For example, at $30^\circ$ (or $\pi/6$), the coordinates are $(\sqrt{3}/2,\,1/2)$, so $\cos30^\circ=\sqrt{3}/2$ and $\sin30^\circ=1/2$. A 30-60-90 right triangle of hypotenuse 1 has legs of length 1/2 and $\sqrt{3}/2$, matching those values. Similarly, a 45-45-90 triangle of hypotenuse 1 has legs of length $1/\sqrt{2}$, giving $\sin45^\circ=\cos45^\circ=1/\sqrt{2}$. This concrete correspondence between unit-circle coordinates and triangle side ratios yields the exact values for common angles.

Remember

Trigonometric identities like $\sin^2(\theta) + \cos^2(\theta) = 1$ are fundamental to solving problems.

Quick Reference

Sine of 30°:$\frac{1}{2}$

Cosine of 60°:$\frac{1}{2}$

Tangent of 45°:$1$

Understanding Values of Trigonometric Ratios for Common Angles

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Beginner

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Beginner Explanation

Beginner Explanation: Draw a right triangle with angle $\theta$. SOH-CAH-TOA tells us $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$. In a 45°-45°-90° triangle the legs are equal and the hypotenuse is $\sqrt{2}$ times a leg, so $\sin45^\circ=\cos45^\circ=1/\sqrt{2}$ and $\tan45^\circ=1$. In a 30°-60°-90° triangle sides are in the ratio 1:√3:2, giving $\sin30^\circ=1/2$, $\cos30^\circ=\sqrt{3}/2$, $\sin60^\circ=\sqrt{3}/2$, $\cos60^\circ=1/2$, and $\tan60^\circ=\sqrt{3}$.

Practice Problems

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Quick Quiz

Single Choice Quiz

Beginner

What is the sine of $30^\circ$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are flying a kite at an angle of $45^\circ$ with the ground. How high is the kite if the string is $20$ meters long?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If $\cos(\theta) = \frac{1}{2}$, find $\theta$ in degrees for $0^\circ \leq \theta < 360^\circ$.

Challenge Quiz

Single Choice Quiz

Advanced

What is the tangent of $60^\circ$?

Recap

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