# 2.1.1: Right Triangle Trigonometry

- Page ID
- 4213

Sine, cosine, tangent, and other ratios of sides of a right triangle.

**Sine**, Cosine, and Tangent

**Sine**, Cosine, and Tangent

** Trigonometry** is the study of the relationships between the sides and angles of right triangles. The legs are called

**or**

*adjacent***depending on which**

**opposite****acute angle**is being used.

\(\begin{aligned} a \text{ is adjacent to } \angle B \qquad a \text{ is opposite } \angle A \\ b \text{ is adjacent to } \angle A \qquad b \text{ is opposite } \angle B\\ c \text{ is the hypotenuse }\end{aligned}\)

The three basic **trigonometric ratios** are called sine, cosine and tangent. For right triangle △ABC, we have:

\(\begin{aligned}

\text { sine Ratio: } \dfrac{\text {opposite leg}}{\text {hypotenuse}} \qquad \sin A=\dfrac{a}{c} \text { or } \sin B=\dfrac{b}{c}\\

\text { cosine Ratio: } \dfrac{\text {adjacent leg}}{\text {hypotenuse}} \qquad \cos A=\dfrac{b}{c} \text { or } \cos B=\dfrac{a}{c}\\

\text { Tangent Ratio: } \dfrac{\text {opposite leg}}{\text {adjacent leg}} \qquad \tan A=\dfrac{a}{b} \text { or } \tan B=\dfrac{b}{a}

\end{aligned}\)

An easy way to remember ratios is to use SOH-CAH-TOA.

**A few important points:**

- Always
(fractions) when you can.**reduce ratios** - Use the
**Pythagorean Theorem**to find the missing side (if there is one). - If there is a
**radical**in the denominator,**rationalize the denominator.**

What if you were given a right triangle and told that its sides measure 3, 4, and 5 inches? How could you find the sine, cosine, and tangent of one of the triangle's non-right angles?

Example \(\PageIndex{1}\)

Find the sine, cosine and tangent ratios of \(\angle A\).

**Solution**

First, we need to use the Pythagorean Theorem to find the length of the **hypotenuse**.

\(\begin{aligned}

5^{2}+12^{2} &=c^{2} \\

13 &=c \end{aligned}\)

\(\begin{aligned}

\sin A &=\dfrac{l e g \text { opposite } \angle A}{\text {hypotenuse}}=\dfrac{12}{13} & \cos A=\dfrac{\text {leg adjacent to } \angle A}{\text {hypotenuse}}=\dfrac{5}{13}, \\

\tan A &=\dfrac{\text {leg opposite } \angle A}{\text {leg adjacent to } \angle A}=\dfrac{12}{5}

\end{aligned}\)

Example \(\PageIndex{2}\)

Find the sine, cosine, and tangent of \(\angle B\).

Find the length of the missing side.

**Solution**

\(\begin{aligned}

A C^{2}+5^{2} &=15^{2} \\

A C^{2} &=200 \\

A C &=10 \sqrt{2} \end{aligned}\)

\(\begin{aligned}\sin B &=\dfrac{10 \sqrt{2}}{15}=\dfrac{2 \sqrt{2}}{3} \quad \cos B=\dfrac{5}{15}=\dfrac{1}{3} \quad \tan B=\dfrac{10 \sqrt{2}}{5}=2 \sqrt{2}

\end{aligned}\)

Example \(\PageIndex{3}\)

Find the sine, cosine and tangent of \(30^{\circ}\).

**Solution**

This is a 30-60-90 triangle. The short leg is 6, \(y=6\sqrt{3}\) and \(x=12\).

\(\sin 30^{\circ}=\dfrac{6}{12}=\dfrac{1}{2} \qquad \cos30^{\circ}=\dfrac{6\sqrt{3}}{12}=\dfrac{\sqrt{3}}{2} \qquad \tan 30^{\circ}=\dfrac{6}{6\sqrt{3}}=\dfrac{1}{\sqrt{3}}\cdot \dfrac{\sqrt{3}}{\sqrt{3}}=\dfrac{\sqrt{3}}{3}\)

Example \(\PageIndex{4}\)

Answer the questions about the following image. Reduce all fractions.

What is \(\sin A\), \(\cos A\), and \(\tan A\)?

**Solution**

\(\begin{array}{l}

\sin A=\dfrac{16}{20}=\dfrac{4}{5} \\

\cos A=\dfrac{12}{20}=\dfrac{3}{5} \\

\tan A=\dfrac{16}{12}=\dfrac{4}{3}

\end{array}\)

**Review**

Use the diagram to fill in the blanks below.

- \(\tan D=\dfrac{?}{?}\)
- \(\sin F=\dfrac{?}{?}\)
- \(\tan F=\dfrac{?}{?}\)
- \(\cos F=\dfrac{?}{?}\)
- \(\sin D=\dfrac{?}{?}\)
- \(\cos D=\dfrac{?}{?}\)

From questions 1-6, we can conclude the following. Fill in the blanks.

- \(\cos \underline{\qquad}=\sin F\) and \(\sin \underline{\qquad}=\cos F\).
- \(\tan D\) and \(\tan F\) are _________ of each other.

Find the sine, cosine and tangent of \(\angle A\). Reduce all fractions and radicals.

**Review (Answers)**

To see the Review answers, open this PDF file and look for section 8.7.

## Resources

## Vocabulary

Term | Definition |
---|---|

Acute Angle |
An acute angle is an angle with a measure of less than 90 degrees. |

Adjacent Angles |
Two angles are adjacent if they share a side and vertex. The word 'adjacent' means 'beside' or 'next-to'. |

Hypotenuse |
The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle. |

Legs of a Right Triangle |
The legs of a right triangle are the two shorter sides of the right triangle. Legs are adjacent to the right angle. |

opposite |
The opposite of a number \(x\) is \(−x\). A number and its opposite always sum to zero. |

Pythagorean Theorem |
The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by \(a^2+b^2=c^2\), where a and b are legs of the triangle and c is the hypotenuse of the triangle. |

Radical |
The \(\sqrt\), or square root, sign. |

sine |
The sine of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the hypotenuse. |

Trigonometric Ratios |
Ratios that help us to unders\tan d the relationships between sides and angles of right triangles. |

## Additional Resources

Interactive Element

Video: Introduction to Trigonometric Functions U\sin g Triangles

Activities: Sine, cosine, Tangent Discussion Questions

Study Aids: Trigonometric Ratios Study Guide

Practice: Right Triangle Trigonometry

Real World: Sine cosine Tangent