By Sky Smith
Updated Aug 30, 2022
Trigonometry explores the relationships between the angles and sides of triangles. In a right triangle, the sine function (sin) connects the length of the side opposite an acute angle to the hypotenuse. By measuring this ratio, you can reverse‑engineer the exact angle that produced it.
Right triangles contain one 90° angle and two acute angles (<90°). Choose the acute angle you want to find; this determines which side is considered opposite it.
Typically you’ll have at least two side lengths. If one side is missing, use the Pythagorean Theorem: a² + b² = c², where c is the hypotenuse. For example, with an adjacent side of 3 and a hypotenuse of 5, compute √(5² – 3²) = √(25 – 9) = √16 = 4 to find the opposite side.
Divide the opposite side by the hypotenuse. Using the previous example, 4 ÷ 5 = 0.8.
Enter the ratio (0.8) into a calculator and use the sin⁻¹ (inverse sine) function to retrieve the angle. The result is approximately 53.13°. The remaining acute angle can be found by subtracting this value from 90°, yielding 36.87°. Together with the right angle, the triangle’s angles are 36.87°, 53.13°, and 90°.
