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Learn Index · Trigonometry Grade 9–10

Solving for unknown sides and angles

Use trig ratios and the Pythagorean theorem to find missing parts.

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What "solving" means

Solving a triangle means finding every unknown side and angle. In a right triangle, you need at least two pieces of information besides the right angle, and at least one must be a side.

Toolkit

Tool Use when
Angle sum you know two angles
Pythagorean theorem you know two sides
Trigonometric ratio you know an angle and a side
Inverse trigonometry you know two sides and want an angle

Case 1: One side and one acute angle

Example: In △ABC, ∠C = 90°, ∠A = 32°, c (hypotenuse) = 20.

  1. ∠B = 90° − 32° = 58°
  2. a (opposite A) = 20 sin 32° ≈ 10.60
  3. b (adjacent A) = 20 cos 32° ≈ 16.96
  4. Check: 10.60² + 16.96² ≈ 400 ✓

Case 2: Two sides

Example: ∠C = 90°, a = 7, b = 24.

  1. c = √(49 + 576) = 25
  2. ∠A = tan⁻¹(7/24) ≈ 16.3°
  3. ∠B = 90° − 16.3° = 73.7°

Strategy tips

  • Draw and label a diagram first.
  • Use given values rather than calculated ones whenever possible, to avoid passing on rounding errors.
  • Round only final answers (usually to one decimal place for angles and lengths, unless told otherwise).
  • Check: the largest angle is opposite the longest side.

Two-triangle problems

Sometimes you must solve one triangle to get information for another.

Example: From a point 50 m from the base of a building, the angle to the top is 38°, and to the top of an antenna on the roof is 44°. Height of the antenna?

  • Building: 50 tan 38° ≈ 39.06 m
  • Building + antenna: 50 tan 44° ≈ 48.28 m
  • Antenna ≈ 9.2 m

Worked examples

Example 1: ∠C = 90°, ∠B = 45°, a = 6. Solve.

∠A = 45°, b = 6, c = 6√2 ≈ 8.49.

Example 2: ∠C = 90°, a = 9, c = 15. Solve.

b = 12; ∠A = sin⁻¹(9/15) ≈ 36.9°; ∠B ≈ 53.1°.

Example 3: A kite string 80 m long makes a 55° angle with the ground. Height of the kite? 80 sin 55° ≈ 65.5 m.

Common mistakes

  • Using rounded values in later steps, causing errors to grow.
  • Mislabelling sides relative to the angle.
  • Forgetting to find all parts of the triangle.

Try it yourself

  1. ∠C = 90°, ∠A = 40°, b = 10. Find a and c.
  2. ∠C = 90°, a = 8, b = 15. Find c, ∠A and ∠B.
  3. ∠C = 90°, ∠B = 62°, c = 30. Find a and b.

Answers: 1) a ≈ 8.39, c ≈ 13.05 2) c = 17, ∠A ≈ 28.1°, ∠B ≈ 61.9° 3) a ≈ 14.08, b ≈ 26.49

Quick recap

  • Solving a triangle means finding every side and angle.
  • Use the angle sum, the Pythagorean theorem, trigonometric ratios and inverse trigonometry.
  • Keep full precision and round only at the end.