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Trigonometric Functions

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Trigonometric Functions

Study Guide

Key Definition

The trigonometric functions are based on ratios of sides in right triangles and can be extended to functions of angles. These include $\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$.

Important Notes

Trigonometric functions are periodic with a period of $2\pi$ radians.
The range of $\sin(\theta)$ and $\cos(\theta)$ is $-1 \leq y \leq 1$.
The domain of $\tan(\theta)$ excludes angles where $\cos(\theta) = 0$.
Radians are often used in trigonometry because they relate directly to the unit circle.
Understanding the unit circle is crucial for mastering trigonometric functions.

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 the division of a by b

Remember to use proper notation when solving problems

Why It Works

Trigonometric functions arise from the unit circle, where angles are measured in radians and the coordinates of points on the circle correspond to $\cos(\theta)$ and $\sin(\theta)$.

Remember

The key relationships are $\sin^2(\theta) + \cos^2(\theta) = 1$ and $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$.

Quick Reference

Sine Function:$y = \sin(x)$

Cosine Function:$y = \cos(x)$

Tangent Function:$y = \tan(x)$

Understanding Trigonometric Functions

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

In a right triangle with angle θ, $\sin(\theta)$ is the opposite side over hypotenuse and $\cos(\theta)$ is adjacent over hypotenuse. On the unit circle, they are the y- and x-coordinates, respectively.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the value of $\sin(\frac{\pi}{2})$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboarder is making a jump off a ramp that forms a $30^\circ$ angle with the ground. Calculate the ratio of the height to the length of the ramp using $\tan(30^\circ)$.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Prove that $\sin^2(\theta) + \cos^2(\theta) = 1$ using the unit circle.

Challenge Quiz

Single Choice Quiz

Advanced

If $\tan(\theta) = \frac{3}{4}$ and θ is acute, what is $\sin(\theta)$?

Recap

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Review key concepts and takeaways