Definition

Sine, cosine and tangent in a right triangle

Trigonometry • Triangle

Statement

In a right triangle with acute angle

θ

sin θ = \frac{opposite}{hypotenuse}

cos θ = \frac{adjacent}{hypotenuse}

and

tan θ = \frac{opposite}{adjacent}

, where the hypotenuse is the side facing the right angle and 'opposite' and 'adjacent' are named relative to

θ

itself.

How to use it

1Name the sides relative to the angle you are using, not relative to the picture: one and the same leg is 'opposite' for one acute angle and 'adjacent' for the other.
2Choose the ratio containing the side you want and the side you know; $tan θ$ is the one to reach for whenever the hypotenuse is neither.
3A known angle and one known side determine the other two sides, which is what makes a single sighting enough in a heights-and-distances problem.

In a right triangle the side opposite

θ

5

and the side adjacent to

θ

12

. Find

tan θ

and

sin θ

Step-by-step

1
$tan θ = \frac{5}{12}$
Why: The two named sides are exactly the pair the tangent uses.
2
$hypotenuse = 5^{2} + 1 2^{2} = 13$
Why: The sine needs the hypotenuse, which Pythagoras supplies.
3
$sin θ = \frac{5}{13}$

Answer

tan θ = \frac{5}{12}

and

sin θ = \frac{5}{13}

Watch out

These ratios are defined for the ACUTE angles of a right triangle; taking the sides from the wrong vertex silently interchanges

sin

with

cos

and

tan

with

cot

. The hypotenuse is never the 'adjacent' side, even though it does touch the angle.

Builds on

Also known as

SOH CAH TOA, trigonometric ratios in a right triangle, opposite over adjacent, definition of sine cosine and tangent, right triangle ratios