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Learn Index · Trigonometry Grade 11–12

Solving non-right triangles

Finding all unknown sides and angles of any triangle.

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Choosing the right tool

To solve a triangle that has no right angle (an oblique triangle), choose between the sine law and the cosine law based on what’s given.

Given First step Then
AAS or ASA sine law angle sum, sine law
SSA sine law (check ambiguous case) angle sum, sine law
SAS cosine law for the third side sine law or cosine law for an angle
SSS cosine law for the largest angle sine law, angle sum

Remember: the three angles always add to 180°.

Example: SAS

b = 9, c = 14, A = 48°

  1. Cosine law: a² = 81 + 196 − 252 cos 48° ≈ 277 − 168.6 = 108.4 → a ≈ 10.41
  2. B is opposite the shorter of b and c, so it’s acute. Sine law: sin B = 9 sin 48°/10.41 ≈ 0.6425 → B ≈ 40.0°
  3. C = 180 − 48 − 40.0 = 92.0°

Example: SSS

a = 6, b = 8, c = 11

  1. Largest angle C: cos C = (36 + 64 − 121)/96 ≈ −0.219 → C ≈ 102.6°
  2. Sine law: sin A = 6 sin 102.6°/11 ≈ 0.532 → A ≈ 32.2°
  3. B = 180 − 102.6 − 32.2 = 45.2°

Area of any triangle

Area = ½ab sin C (two sides and the included angle)

From the SAS example: ½(9)(14) sin 48° ≈ 46.8 square units.

When you know all three sides, Heron’s formula also works: A = √[s(s − a)(s − b)(s − c)], where s is half the perimeter.

Bearings

A bearing is a direction measured clockwise from north (000° to 360°).

Example: A plane flies 200 km on a bearing of 060°, then 150 km on a bearing of 150°. How far is it from the start?

  • The turn between the legs: 150° − 60° = 90° → the angle inside the triangle is 90°
  • d = √(200² + 150²) = 250 km

(If the angle isn’t 90°, use the cosine law.)

Real-world applications

  • Surveying land with inaccessible sides
  • Navigation for ships and planes
  • Engineering trusses and roof frames
  • Finding distances across lakes or canyons

Worked examples

Example 1: A = 35°, B = 75°, c = 20. Find a. C = 70° → a = 20 sin 35°/sin 70° ≈ 12.2.

Example 2: Area with a = 7, b = 10, C = 30°: ½(7)(10)(0.5) = 17.5.

Example 3: Which law first for a = 5, b = 7, c = 9? Cosine law (SSS).

Common mistakes

  • Using the sine law when you don’t have a matching pair.
  • Rounding intermediate values too much.
  • Forgetting the ambiguous case with SSA.

Try it yourself

  1. Which law for A = 40°, b = 12, c = 15?
  2. Solve: A = 40°, b = 12, c = 15 (find a).
  3. Area of the triangle in question 2.
  4. A ship sails 30 km north, then 40 km on a bearing of 120°. Angle at the turn inside the triangle?

Answers: 1) cosine law (SAS) 2) a ≈ 9.65 3) ≈ 57.9 square units 4) 60°

Quick recap

  • Pick the sine law or cosine law based on the given information.
  • Area formula: ½ab sin C.
  • Bearings are measured clockwise from north.