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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 rootRemember 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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concept. Use space or enter to play video.concept thumbnailBeginner

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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: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

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

A$\frac{1}{2}$B$\frac{\sqrt{3}}{2}$C$\frac{1}{\sqrt{2}}$D$1$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

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?Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

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

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

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

A$\frac{3}{5}$B$\frac{4}{5}$C$\frac{5}{4}$D$\frac{1}{5}$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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