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Sine Function

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Sine Function

Study Guide

Key Definition

The sine function is defined for an angle $\theta$ as $\sin(\theta)$ which is the y-coordinate of the point on the unit circle.

Important Notes

The sine function is periodic with a period of $2\pi$
The amplitude of $\sin(x)$ is 1
Key angles to remember: $\sin(0^\circ) = 0$, $\sin(90^\circ) = 1$
$\sin(\theta)$ is positive in the first and second quadrants
$\sin(\theta)$ is negative in the third and fourth quadrants

Mathematical Notation

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

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

$\pi$ represents the mathematical constant pi

$\frac{x}{y}$ represents division

$\sqrt{x}$ represents the square root

Remember to use proper notation when solving problems

Why It Works

The sine function works as it represents the y-coordinate on the unit circle, a fundamental concept in trigonometry.

Remember

The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

Quick Reference

Period:$2\pi$

Amplitude:<a href="amplitude-and-period-of-sine-and-cosine-functions">$1$</a>

Key Value:$\sin(90^\circ) = 1$

Understanding Sine Function

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The sine function is a periodic function that repeats every $2\pi$, and it measures the y-coordinate on the unit circle.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is $\sin(90^\circ)$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A Ferris wheel with radius 10 m is centered 10 m above the ground. Calculate the height above the ground when it rotates $30^\circ$ from the lowest point (angle measured from the lowest point).

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Determine the period of the function $y = 3\sin(x)$.

Challenge Quiz

Single Choice Quiz

Advanced

What is the value of $\sin(\pi)$?

Recap

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Review key concepts and takeaways