By Michael O. Smathers – Updated Aug 30, 2022
Trigonometry is the branch of mathematics that focuses on the relationships between a triangle’s sides and angles. It is typically taught at the pre‑calculus or advanced geometry level and is essential for solving problems where only partial information about a triangle is known. In this guide, we’ll walk through the most reliable methods for determining unknown angles when you have two sides, one side and an angle, or no right angles at all.
For a right triangle, input the two known legs (A and B) into the classic equation:
A2 + B2 = C2 – where C is the hypotenuse, the side opposite the right angle. For example, if the legs measure 3 cm and 4 cm, calculate 3² + 4² = 9 + 16 = 25.
Subtract the square of the known side from the square of the hypotenuse to isolate the missing side’s square. If the known leg is 5 cm and the hypotenuse is 13 cm, subtract 5² = 25 from 13² = 169, yielding 144.
Take the square root of the result to find the length of the missing leg. The square root of 144 is 12, so the other side measures 12 cm.
Select the angle adjacent to the known hypotenuse and one leg. This angle will be used to compute the sine and cosine values.
Calculate the sine by dividing the side opposite the chosen angle by the hypotenuse. For a triangle with a hypotenuse of 13 cm and an opposite side of 12 cm, sin θ = 12/13 ≈ 0.923.
Calculate the cosine by dividing the side adjacent to the chosen angle by the hypotenuse. Using the same triangle, cos θ = 5/13 ≈ 0.384.
On a scientific calculator, input the sine or cosine value and press the inverse (inv) button to retrieve the angle. For sin 0.923 or cos 0.384, the result is approximately 67.38°.
To find the remaining angle, add 90° to the calculated angle and subtract the sum from 180°. In this example: 67.38° + 90° = 157.38°, and 180° – 157.38° = 22.62°.
When the triangle has no right angle, the Law of Sines provides a powerful tool for relating angles to opposite sides. According to Clark University, the law is expressed as:
sin a / A = sin b / B = sin c / C, where a, b, c are angles and A, B, C are the sides opposite them.
Identify one known side and its opposite angle. If the angle is obtuse (between 90° and 180°), first find the supplementary acute angle by subtracting from 180°.
Compute the sine of the known angle by dividing the opposite side by the adjacent side, or by using a calculator if only the side lengths are given.
Set the ratio sin a / A equal to sin b / B (or sin c / C) and solve for the unknown sine value. Multiply both sides by the corresponding side length to isolate the sine of the unknown angle.
Use the inverse sine function on a calculator to convert the sine value back to an angle. Repeat the process for each remaining angle.
By following these systematic steps, you can confidently determine all unknown angles of any triangle, whether it contains a right angle or not.
