Trigonometric Values for Common Angles

Study Guide

Key Definition

Trigonometric functions like $\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$ help calculate relationships in right triangles.

Important Notes

The sine of 30^\circ is $\frac{1}{2}$
The cosine of 45^\circ is $\frac{\sqrt{2}}{2}$
The tangent of 60^\circ is $\sqrt{3}$
The secant is the reciprocal of cosine
The cosecant is the reciprocal of sine

Mathematical Notation

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

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

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

$\csc(\theta)$ is the cosecant function

$\sec(\theta)$ is the secant function

$\cot(\theta)$ is the cotangent function

Remember to use proper notation when solving problems

Why It Works

Trigonometric values can be derived from special triangles and the unit circle. On a unit circle of radius 1, a point at angle θ has coordinates (cos θ, sin θ), giving $\sin(θ)=\frac{\text{opposite}}{\text{hypotenuse}}$ and $\cos(θ)=\frac{\text{adjacent}}{\text{hypotenuse}}$. For example, in a 30°–60°–90° triangle with sides 1, √3, and 2, you obtain $\sin(30^\circ)=\frac{1}{2}$, $\cos(30^\circ)=\frac{\sqrt{3}}{2}$, illustrating how the ratios remain constant for a given angle.

Remember

Each trigonometric function has a specific ratio: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

Quick Reference

Sine 30^\circ:$\frac{1}{2}$

Cosine 45^\circ:$\frac{\sqrt{2}}{2}$

Tangent 60^\circ:$\sqrt{3}$

Understanding Trigonometric Values for Common Angles

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Video explanation of this concept

Beginner

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Beginner Explanation

Use special right triangles: in a 30°–60°–90° triangle with sides 1, √3, and 2, you get $\sin(30^\circ)=\frac{1}{2}$, $\cos(30^\circ)=\frac{\sqrt{3}}{2}$, $\tan(30^\circ)=\frac{1}{\sqrt{3}}$. In a 45°–45°–90° triangle with legs of length 1 and hypotenuse √2, you get $\sin(45^\circ)=\cos(45^\circ)=\frac{\sqrt{2}}{2}$ and $\tan(45^\circ)=1$.

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 if the angle of elevation is 45^\circ and the distance to the tree is 10 meters.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Find the value of $\cos(60^\circ)$ using a trigonometric identity.

Challenge Quiz

Single Choice Quiz

Advanced

What is $\tan(45^\circ)$?

Recap

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Review key concepts and takeaways