Calculating an unknown length

The 4-step method

To calculate an unknown side length in a right-angled triangle:

1. Locate the known angle and the right angle.
2. Identify the relevant sides (opposite, adjacent, hypotenuse) in relation to that angle.
3. Choose the correct formula (sin, cos or tan) depending on the sides you know and the one you are looking for.
4. Isolate the unknown and calculate it using a calculator (in degrees mode!).

Example

A right-angled triangle has an angle of 30 degrees and a hypotenuse of 8 cm. We are looking for the side opposite this angle. We know the hypotenuse and are looking for the opposite side -> we use the sine function. sin(30) = opposite / 8 opposite = 8 × sin(30) = 8 × 0.5 = 4 cm

Another example

A right-angled triangle has an angle of 50 degrees and an adjacent side of 6 cm. We are looking for the hypotenuse. cos(50) = 6 / hypotenuse hypotenuse = 6 / cos(50) = 6 / 0.643 ≈ 9.3 cm

Common pitfalls

• Check that your calculator is set to DEG (degrees) mode and not RAD (radians), otherwise all the results will be incorrect.
• When the unknown is in the denominator (as with the hypotenuse above), you divide rather than multiply: don’t skip this step of isolating the unknown.