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MathidiaMathematics Academy
Trigonometry and bearings
Lesson 52 of 75
Easy practice

Missing sides

Objective. Rearrange trigonometric equations to calculate lengths.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
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  5. 5. Continue

Learn: the key idea

Missing sides in right-angled triangles are found by writing a trig equation from the known angle and side, then rearranging to isolate the unknown. Set up as sin θ = opp/hyp (or cos/tan equivalents), substitute known values, then multiply or divide to solve. For example, with angle 30° and hypotenuse 8 cm, opposite = 8 × sin 30° = 4 cm. If the unknown is on the bottom of the fraction, you divide the known side by the trig ratio instead of multiplying.

Worked example

A right-angled triangle has angle 50° and adjacent side 6 cm. Find the length of the hypotenuse to find this missing side, to 1 decimal place.

Diagram of a triangle
  1. 1cos 50° = adjacent/hypotenuse = 6/h
  2. 2Rearrange: h = 6/cos 50°
  3. 3Compute cos 50° ≈ 0.6428
  4. 4h ≈ 6/0.6428 ≈ 9.3 cm

Answer: 9.3 cm

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Common mistake. A common error is multiplying by the trig ratio when the unknown is actually in the denominator; always rearrange algebraically first rather than guessing whether to multiply or divide.

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