# Tangent: in a Right Triangle

How to find sine in a right triangle (trigonometry): formula, meaning, 3 examples, and their solutions.

## Formula

### Formula

Tangent is the ratio of

[Opposite side]/[Adjacent]

in a right triangle.

The opposite side means

the side opposite to ∠A.

The adjacent side means

the side adjancent to ∠A

(which is not the hypotenuse).

To remember the ratio,

remember TOA:

Tangent, Opposite side, and Adjacent side.

## Meaning

### Meaning

tan A

= [Opposite side]/[Adjacent side]

= [change of y]/[change of x]

The slope of a line is

m

= [y_{2} - y_{1}]/[x_{2} - x_{1}]

= [change of y]/[change of x].

So tangent means

the slope of a right triangle.

## Example 1

### Example

### Solution

Tangent is TOA:

Tangent, Opposite side, and Adjacent side.

The Opposite side is 3.

The Adjacent side is 4.

So,

T, tan A

is equal to,

O: opposite side, 3

over,

A: adjacent side, 4.

So tan A = 3/4.

## Example 2

### Example

### Solution

Tangent is TOA:

Tangent, Opposite side, and Adjacent side.

The Opposite side is 12.

The Adjacent side is 5.

So,

T, tan A

is equal to,

O: opposite side, 12

over,

A: adjacent side, 5.

So tan A = 12/5.

## Example 3

### Example

### Solution

First, find tan A

from the given right triangle.

Tangent is TOA:

Tangent, Opposite side, and Adjacent side.

The Opposite side is x.

The Adjacent side is 12.

So,

T, tan A

is equal to,

O: opposite side, x

over,

A: adjacent side, 12.

Next, it says

tan A = 7/6.

So write

[ = 7/6].

So tan A = x/12 = 7/6.

Solve x/12 = 7/6.

Multiply 12 to both sides.

Then x = [7/6]⋅12.

Cancel the denominator 6

and reduce the numerator 12 to, 12/6, 2.

7⋅2 = 14

So x = 14.