Calculate an unknown angle

Finding an angle using reciprocal functions

When you know two sides of a right-angled triangle but not the angle, you use the reciprocal functions: sin⁻¹, cos⁻¹ and tan⁻¹ (often denoted arcsin, arccos and arctan). On a calculator, these are usually accessed by pressing the shift or 2nd key followed by sin, cos or tan.

Example

A right-angled triangle has an opposite side of 5 cm and an adjacent side of 12 cm (a 5-12-13 triangle). We are looking for the angle x formed with the adjacent side. We know the opposite and adjacent sides -> we use the tangent. tan(x) = 5/12 = 0.4167 x = tan⁻¹(0.4167) ≈ 22.6 degrees

Verification using another ratio

We can verify this using the hypotenuse (13 cm): sin(x) = 5/13 = 0.3846 → x = sin⁻¹(0.3846) ≈ 22.6 degrees. Both methods give the same result, which is reassuring!

Common pitfalls

• Do not confuse sin(x) = ... (gives a ratio) with arcsin(ratio) = ... (gives an angle). These are inverse operations.
• Don’t forget the unit: the result of a reciprocal function is an angle in degrees, not a length.
• Only round off at the very end of the calculation to ensure accuracy.