Cosine Law. Sine Law. Ambiguous Case of Sine Law
Lesson 6: Cosine Law. Sine Law. Ambiguous Case
Solve triangles with the sine and cosine laws, including the ambiguous case.
When we need to determine angle measure or side lengths within the non-right triangles, we use Cosine Law and Sine Law.
Cosine Law is used in two cases: SAS and SSS
- When we are given all three sides, no angles and are looking for an angle
- When we are given 2 sides and a contained angle are looking for the side length that corresponds to the given angle.
Cosine Law Formula(e)
When solving for an angle, the formula needs to be rearranged and the last step is always taking the inverse ratio to get the measure of the angle.
Sine Law is used in cases when Cosine Law cannot be used.
Usually when we have pairs of corresponding sides and angles, within one of which either a side or an angle is unknown: SAA or SSA
Sine Law Formula
Normally, we set up an equation involving two ratios provided and cross-multiply to isolate for the unknown value.
Ambiguous Case of Sine Law
Whenever we are given two sides and an angle and are lookin for another angle, we can use Sine Law. However, we need to be aware of the possibilities that the relationships between the given measurements could results in. Sometimes, the relationship between the sides and an angle do not even produce a triangle. In other cases they may result in two possible triangles and we need to consider both situations. This is called an Ambiguous Case of Sine Law.
Below is a diagram of possible outcomes depending on the measurements given:
Learn more about Cosine Law, Sine Law and the Ambiguous Case of Sine Law and see worked examples by watching the video lesson:
Key ideas: Cosine Law. Sine Law. Ambiguous Case
- Cosine law: a2 = b2 + c2 − 2bc cos A. Use it for SAS (find the third side) or SSS (find an angle: cos A = (b2 + c2 − a2)/(2bc)).
- Sine law: a/sin A = b/sin B = c/sin C. Use it when you know a side and its opposite angle (AAS, ASA, SSA).
- Ambiguous case (SSA): given angle A, side a opposite it, and side b. With h = b sin A: if a < h, no triangle; a = h, one right triangle; h < a < b, two triangles; a ≥ b, one triangle.
Worked examples
Example 1
In △ABC, b = 8, c = 11 and ∠A = 37°. Find a.
- SAS → cosine law: a2 = 82 + 112 − 2(8)(11) cos 37°.
- a2 ≈ 64 + 121 − 140.6 = 44.4.
- a ≈ 6.67
Example 2
A triangle has sides 7, 9 and 12. Find the largest angle.
- The largest angle is opposite the longest side (12): cos C = (72 + 92 − 122) / (2 × 7 × 9) = −14/126.
- C = cos−1(−0.111) ≈ 96.4° (an obtuse angle, since the cosine is negative).
Example 3
In △ABC, ∠A = 40°, ∠B = 65° and a = 10. Find b.
- AAS → sine law: b/sin 65° = 10/sin 40°.
- b = 10 sin 65° / sin 40° ≈ 14.1
Example 4
∠A = 30°, a = 6 and b = 10. How many triangles are possible? Find ∠B.
- h = b sin A = 10 sin 30° = 5. Since 5 < 6 < 10 (h < a < b), there are two triangles.
- sin B = 10 sin 30° / 6 ≈ 0.833, so B ≈ 56.4° or B ≈ 180° − 56.4° = 123.6°.
Common mistakes
- Using the sine law when you do not know any side–opposite angle pair (use the cosine law first).
- Forgetting the second, obtuse solution in the ambiguous case.
- Order of operations in the cosine law: subtract 2bc cos A from b2 + c2, do not multiply (b2 + c2 − 2bc) by cos A.
Questions students ask
How do I know which law to use?
If you know a side and its opposite angle, use the sine law. Otherwise (SAS or SSS), start with the cosine law.
Why does the ambiguous case have two answers?
Because sin θ = sin(180° − θ), the side a can swing to two different positions and still reach the base.
Check your understanding
Practise with the Grade 11 quizzes. You’ll see right away if each answer is correct.