Trigonometric Ratios - Varsity Tutors

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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 xRemember 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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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.Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

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

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

Real-World Problem

Question ExerciseIntermediate

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

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

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

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

Challenge Quiz

Single Choice QuizAdvanced

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

A$\frac{5}{13}$B$\frac{12}{13}$C$\frac{5}{\sqrt{169}}$D$\frac{12}{\sqrt{169}}$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

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