Laws of Sines and Cosines

Grade 10 · Math · Study Guide

Solving non-right triangles using the Law of Sines and Law of Cosines.

SOH-CAH-TOA only works for right triangles. For any other triangle — acute or obtuse — you need the Law of Sines or the Law of Cosines. These two laws are your tools for solving any triangle, given enough information.

The Law of Sines states: a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are the sides of the triangle and A, B, C are the angles opposite those sides. It's ideal when you know two angles and one side (AAS or ASA), or two sides and an angle opposite one of them (SSA). The SSA case is tricky because it can produce zero, one, or two valid triangles — this is known as the 'ambiguous case.'

The Law of Cosines states: c² = a² + b² - 2ab·cos(C). It generalizes the Pythagorean theorem — notice that when C = 90°, cos(90°) = 0 and the formula reduces to c² = a² + b². Use the Law of Cosines when you know two sides and the included angle (SAS), or all three sides (SSS). For SSS, rearrange to find an angle: cos(C) = (a² + b² - c²) / (2ab).

A good strategy: if you know an angle and its opposite side, the Law of Sines is usually simpler. If you don't have that pairing, use the Law of Cosines first. After finding one piece of information, you can often switch to the simpler Law of Sines to finish the problem.

Which Law to Use
Given Information	Use	Case Name
Two angles + any side	Law of Sines	AAS or ASA
Two sides + non-included angle	Law of Sines	SSA (check ambiguous)
Two sides + included angle	Law of Cosines	SAS
Three sides	Law of Cosines	SSS

Key terms

Law of Sines: a/sin(A) = b/sin(B) = c/sin(C); used for AAS, ASA, SSA triangles.
Law of Cosines: c² = a² + b² - 2ab·cos(C); used for SAS and SSS triangles.
Ambiguous Case (SSA): When given two sides and a non-included angle, there may be 0, 1, or 2 valid triangles.
Included Angle: The angle formed between two known sides.
Solving a Triangle: Finding all three sides and all three angles of a triangle.

Flashcards

Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
Law of Cosines: c² = a² + b² - 2ab·cos(C)
Use Law of Sines when...: You know a side and its opposite angle (AAS, ASA, or SSA)
Use Law of Cosines when...: You know SAS or SSS
Ambiguous case: SSA — may produce 0, 1, or 2 triangles depending on the values
Law of Cosines with SSS (finding angle): cos(C) = (a² + b² - c²) / (2ab)

Practice questions

When should you use the Law of Cosines instead of the Law of Sines?
1. When you know two angles and one side
2. When you know two sides and the included angle (SAS)
3. When you know angle-angle-side
4. When you have a right triangle
Show answer
Answer: When you know two sides and the included angle (SAS)
SAS (two sides and their included angle) requires the Law of Cosines. Law of Sines needs an angle paired with its opposite side.
The Law of Sines states: a/sin(A) = b/sin(B) = c/sin(___).
Show answer
Answer: C
The Law of Sines ratio is a/sinA = b/sinB = c/sinC — each side divided by the sine of its opposite angle.
When C = 90°, the Law of Cosines reduces to the Pythagorean theorem.
Show answer
Answer: True
cos(90°)=0, so c²=a²+b²-2ab(0) = a²+b², which is exactly the Pythagorean theorem.
The 'ambiguous case' occurs with which given information?
1. SAS
2. SSS
3. SSA
4. AAS
Show answer
Answer: SSA
SSA (two sides and a non-included angle) is the ambiguous case. The given angle might match 0, 1, or 2 triangles.
The Law of Sines can be used to solve any triangle if you know all three sides (SSS).
Show answer
Answer: False
The Law of Sines requires a side-angle pair (each side with its opposite angle). With SSS you have no angles, so you must use the Law of Cosines.

Fun facts

Surveyors use the Law of Sines to triangulate the positions of distant landmarks — it's how maps and GPS coordinates were determined long before satellites.
The Law of Cosines was known to Islamic mathematicians as early as the 10th century, over 500 years before it appeared in European textbooks.

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