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Trigonometric Ratios

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Trigonometric Ratios

Study Guide

Key Definition

Trigonometric ratios are the ratios of the lengths of sides in a right triangle relative to one of its angles, defined as $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$.

Important Notes

Sine, cosine, and tangent are fundamental trigonometric functions.
Each function relates an angle to two side lengths of a right triangle.
The hypotenuse is always the longest side.
$\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$ are often used in calculations involving right triangles.
Trigonometric ratios are essential in many real-world applications, including navigation and physics.

Mathematical Notation

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

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

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

$\frac{a}{b}$ denotes division of a by b

$\sqrt{x}$ represents the square root of x

Remember to use proper notation when solving problems

Why It Works

Trigonometric ratios provide a consistent way to relate angles and side lengths in right triangles, allowing for calculations that are applicable in various fields, such as engineering and physics.

Remember

To find a trigonometric ratio, identify the reference angle and the corresponding sides (opposite, adjacent, hypotenuse) in the triangle.

Quick Reference

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

Cosine:$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

Tangent:$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Understanding Trigonometric Ratios

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Video explanation of this concept

Beginner

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Beginner Explanation

Trigonometric ratios relate the angles of a right triangle to the lengths of its sides. For an acute angle θ in a right triangle, $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. These definitions provide a foundation for solving problems involving right triangles.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which is the correct ratio for $\sin(\theta)$ in a right triangle?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You are standing 50 meters away from a building. The angle of elevation to the top of the building is $30^\circ$. How tall is the building?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

In a triangle, if $\sin(\theta) = \frac{3}{5}$, what is $\cos(\theta)$?

Challenge Quiz

Single Choice Quiz

Advanced

If $\tan(\theta) = \frac{5}{12}$, which of the following is $\sin(\theta)$?

Recap

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Review key concepts and takeaways