By Doug Leenhouts – Updated Aug 30, 2022

Trigonometry examines the relationships between angles and sides in triangles. By applying a few foundational rules—such as the fact that the sum of the interior angles of any triangle equals 180°—you can calculate an unknown angle without needing a protractor. This guide walks you through the process step by step.

Step 1 – Identify the Triangle Type

Start by determining the category of the triangle containing the unknown angle:

• Equilateral – all three sides and angles are equal.
• Isosceles – two equal sides and two equal angles.
• Right – one 90° angle and two acute angles.
• Scalene (irregular) – all sides and angles differ.

Step 2 – Construct a Right Triangle

Draw a perpendicular from one of the other angles to the opposite side. The resulting right triangle will let you use trigonometric ratios.

Step 3 – Measure the Adjacent Side

In the right triangle, the side adjacent to the unknown angle (but not the hypotenuse) is the “adjacent side.” Measure its length.

Step 4 – Measure the Hypotenuse

Measure the side opposite the right angle—the longest side—known as the “hypotenuse.”

Step 5 – Apply the Inverse Cosine Function

Compute the ratio of the adjacent side to the hypotenuse and then use the inverse cosine (cos⁻¹) on a scientific calculator:

cos⁻¹( adjacent / hypotenuse ) = unknown angle

For example, if the adjacent side is 1 unit and the hypotenuse is 2 units, the calculation is:

cos⁻¹(1/2) = 60°

Tools You’ll Need

• Pencil
• Paper
• Ruler
• Scientific calculator (with inverse cosine capability)

With these steps, you can confidently determine any unknown angle in a triangle, leveraging the power of trigonometry and a scientific calculator.