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
= [y2 - y1]/[x2 - x1]
= [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.