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Sine

Overview of the sine function: definition, properties, and use in right triangle trigonometry.

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Sine

Study Guide

Key Definition

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

Important Notes

The sine ratio is the same regardless of the size of the right triangle.
Often, it's easiest to consider a right triangle with a hypotenuse of length 1.
The reciprocal of the sine ratio is the csc ratio: $\csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}}$.
Sine can be thought of as a function that takes different values depending on the measure of the angle.
You can measure an angle in degrees or radians.

Mathematical Notation

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

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

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

$\sqrt{x}$ represents the square root

Remember to use proper notation when solving problems

Why It Works

The sine function relates the angle to the ratio of specific sides in a right triangle, providing a consistent measure across different sizes of triangles.

Remember

For any right triangle, $\sin(\theta)$ will always equal the length of the opposite side divided by the length of the hypotenuse.

Quick Reference

Sine Function:$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

Cosecant Function:$\csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}}$

Understanding Sine

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Video explanation of this concept

Beginner

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Beginner Explanation

Sine is a basic trigonometric function that helps in finding the ratio of the opposite side to the hypotenuse in a right triangle: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the sine of a 30^{\\circ} angle in a right triangle?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you're flying a kite and the string makes a 45^{\\circ} angle with the ground. If the string length is 10 meters, how high is the kite from the ground?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If $\sin(\theta) = \frac{3}{5}$, find $\csc(\theta)$.

Challenge Quiz

Single Choice Quiz

Advanced

In a right triangle, if $\sin(\theta) = \frac{4}{5}$, what is \cos(\theta)?

Recap

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