Right Triangle - Unknown Angle using Cosine

\[\theta=\cos^{-1}\left(\frac{adjacent}{hypotenuse}\right)\]

Variables

thetaunknown angle of the right triangle (rad or °)

adjacentlength of the adjacent side (m)

hypotenuselength of the hypotenuse (m)

Description

What is this formula?

This trigonometric formula calculates an unknown angle in a right triangle using the inverse cosine function.

The cosine ratio relates the adjacent side of an angle to the hypotenuse. Applying the inverse cosine function allows the angle to be determined from known side lengths.

When to use it

Use this formula when:

- The adjacent side is known

- The hypotenuse is known

- An acute angle must be calculated

- Solving right triangle problems

- Working with horizontal distances or vector components

The resulting angle may be expressed in degrees or radians depending on calculator settings.

Example

If:

adjacent = 8 m

hypotenuse = 10 m

Then:

theta = acos(8/10)

theta = acos(0.8)

theta ≈ 36.87°

Applications

- Trigonometry and geometry

- Engineering calculations

- Physics and mechanics

- Navigation and surveying

- Structural analysis

- Computer graphics