Hypotenuse, Adjacent And Opposite Sides Of A Right Triangle

Sides Of A Right Triangle

Hypotenuse, Adjacent and Opposite Sides.

In the following right triangle PQR,

• the side PQ, which is opposite to the right angle PRQ is called the hypotenuse. The hypotenuse is the longest side of the right triangle.
• the side RQ is called the adjacent side of angle θ.
• the side PR is called the opposite side of angle θ.

Note: The adjacent and the opposite sides depend on the angle θ. For complementary angle of θ, the labels of the 2 sides are reversed.

Example:
Identify the hypotenuse, adjacent side and opposite side in the following triangle:
a) for angle x
b) for angle y

Solution:
a) For angle x: AB is the hypotenuse, AC is the adjacent side , and BC is the opposite side.
b) For angle y: AB is the hypotenuse, BC is the adjacent side , and AC is the opposite side.

SOH-CAH-TOA

The following diagram show how to use SOHCAHTOA for a right triangle.

How to identify the Opposite Sides, Adjacent Sides and Hypotenuse of a Right Triangle?

Definition of Cos, Sin, Tan, Csc, Sec, Cot for the right triangle
sin x = opposite/hypotenuse
cos x = adjacent/hypotenuse
tan x = opposite/adjacent
csc x = 1/sin x = hypotenuse/opposite
sec x = 1/cos x = hypotenuse/adjacent
cot x = 1/tan x = adjacent/opposite

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Using the Sine Formula (the SOH formula)

The first part of this video will explain the difference between the hypotenuse, adjacent and opposite sides of a right triangle. Then it shows how to use the sine formula (the SOH formula).

Sine = Opposite over the Hypotenuse

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Using the Cosine Formula (the CAH formula)

Cosine = Adjacent over Hypotenuse

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Using the Tangent Formula (the TOA formula)

Tangent = Opposite over Adjacent

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