Trigonometric Ratio: Tangent

Trigonometric Ratio: Tangent

How to solve tangent (trigonometric ratio) problems: definition, examples, and their solutions.

Definition

tan A = (Opposite side) / (Adjacent side). Remember: TOA (Tangent, Opposite, and Adjacent).

Tangent is the ratio of a right triangle's
(Opposite side) / (Adjacent side).

TOA:
Tangent = (Opposite) / (Adjacent)

Meaning

Tangent means 'slope'. Compare tangent with the slope of a line. Both ratios mean (vertical change, blue) / (horizontal change, purple).

Tangent means 'slope'.

Compare tangent with the slope of a line.

Both ratios mean
(vertical change, blue) / (horizontal change, purple).

Example 1

Find tan A. Hypotenuse: 5, The side opposite to angle A: 3, The side adjacent to angle A: 4.

Tangent: TOA.
So tan A = (Opposite side, 3) / (Adjacent side, 4).

Example 2

Find tan A. Hypotenuse: 13, The side opposite to angle A: 12, The side adjacent to angle A: 5.

Tangent: TOA.
So tan A = (Opposite side, 12) / (Adjacent side, 5).

Example 3

If tan A = 12/5, find the value of x. The side opposite to angle A: x, The side adjacent to angle A: 10.

Tangent: TOA.
So tan A = (Opposite side, x) / (Adjacent side, 10) = 12/5.