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§ Trigonometry

Sine & Cosine Rules

CCSS.HSG.SRT3 min read

The sine and cosine rules unlock triangle problems that basic right-triangle trigonometry cannot solve. These laws help students find missing sides and angles in any triangle, making them essential tools for CCSS.HSG.SRT geometry standards.

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§ 01

Why it matters

Engineering students use these rules to calculate structural loads in non-right triangles, while surveyors apply them to measure distances across irregular terrain. In navigation, pilots use the cosine rule to determine flight paths between 3 airports forming a triangle. Architecture students calculate roof angles and support beam lengths using both rules. A surveyor measuring a triangular plot with sides 150m, 200m, and an included angle of 75° needs the cosine rule to find the third side (approximately 235m). GPS systems rely on triangulation principles that mirror these fundamental laws, processing millions of triangle calculations daily to pinpoint locations within 3 meters of accuracy.

§ 02

How to solve sine & cosine rules

Sine & Cosine Rules

  • Law of sines: a/sin(A) = b/sin(B) = c/sin(C). Use for AAS or SSA.
  • Law of cosines: c² = a² + b² − 2ab·cos(C). Use for SAS (find third side).
  • Rearranged: cos(C) = (a² + b² − c²)/(2ab). Use for SSS (find an angle).
  • Each side is paired with the sine of the angle opposite it.

Example: a=5, b=7, C=60° → c² = 25 + 49 − 70·(12) = 39, so c ≈ 6.24.

§ 03

Worked examples

Beginner§ 01

You are given two sides and the included angle of a triangle. Which rule applies, and what is its formula?

Answer: Law of cosines: c² = a² + b² − 2ab·cos(C)

  1. Recognise the SAS configuration → Scenario: SAS — AAS / SSA → sine rule. SAS / SSS → cosine rule.
  2. Write the formula → c² = a² + b² − 2ab·cos(C) — Use the law of cosines when this configuration is given.
Easy§ 02

In a triangle, side a = 10, angle A = 45°, angle B = 60°. Find side b.

Answer: b ≈ 12.25

  1. Identify the rule → AAS → law of sines — With two angles and a non-included side (AAS), the law of sines applies.
  2. Write the formula with given values → 10/sin(45°) = b/sin(60°) — Pair each side with the sine of its opposite angle.
  3. Solve for b → b = 10 · sin(60°) / sin(45°) = 10 · 0.866 / 0.7071 — Multiply both sides by sin(B) to isolate b.
  4. Approximate to 2 decimals → b ≈ 12.25 — Evaluate numerically to the requested precision.
Medium§ 03

In a triangle, side a = 8, side b = 5, and the included angle C = 60°. Find side c.

Answer: c ≈ 7.0

  1. Identify the rule → SAS → law of cosines — Two sides and the included angle → use the law of cosines.
  2. Write the formula with given values → c² = 8² + 5² − 2·8·5·cos(60°) — c² = a² + b² − 2ab·cos(C).
  3. Solve algebraically → c² = 64 + 25 − 80·0.5 = 49.0 — Compute each term, then combine.
  4. Take square root and round → c = √49.0 ≈ 7.0 — Side lengths are positive; round to 2 decimals.
§ 04

Common mistakes

  • Students confuse when to use each rule, applying sine rule to SAS problems like finding the third side when given a=6, b=8, C=45°, incorrectly getting 6/sin(A) = 8/sin(B) instead of using c² = 36 + 64 - 96cos(45°) = 32.2
  • When using law of cosines to find angles, students forget to use the rearranged formula cos(C) = (a²+b²-c²)/(2ab), instead trying c² = a² + b² - 2ab·cos(C) and getting stuck with the unknown angle
  • Students mix up opposite sides and angles in sine rule, writing a/sin(B) = b/sin(A) instead of a/sin(A) = b/sin(B), leading to incorrect ratios like 10/sin(60°) = 8/sin(30°) giving 11.5 instead of 4.6
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§ 05

Frequently asked questions

How do I know whether to use sine rule or cosine rule?
Use sine rule for AAS (two angles, one side) or ASA configurations. Use cosine rule for SAS (two sides, included angle) or SSS (all three sides). The key is identifying what information you have: if you know two angles, use sine rule; if you know two or three sides, use cosine rule.
Why does the cosine rule work for any triangle type?
The cosine rule reduces to the Pythagorean theorem when the angle is 90°, since cos(90°) = 0. For acute triangles, the cosine term subtracts from a² + b², while for obtuse triangles, the negative cosine adds to the sum, accounting for the triangle's shape in all cases.
Can I use sine rule when I have SSA (two sides, one non-included angle)?
Yes, but be careful of the ambiguous case. With SSA, you might get 0, 1, or 2 valid triangles. Check if the given side opposite the known angle is longer than the other given side to avoid impossible triangles or multiple solutions.
What's the most efficient way to find all missing parts of a triangle?
Start with what you have. For SAS or SSS, use cosine rule first. For AAS or ASA, use sine rule. After finding one missing piece, you often have enough information to use the simpler sine rule for remaining unknowns, since angles in a triangle sum to 180°.
Why do calculator answers sometimes seem wrong for these problems?
Ensure your calculator is in the correct mode (degrees vs radians) and that you're using the right formula variation. When finding angles with inverse cosine, remember that cos⁻¹ only gives angles from 0° to 180°, which works perfectly for triangle angles but can confuse students expecting negative results.
§ 06

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