Trigonometry
Sine Rule (Law of Sines)
Solve any triangle when you know angle-side pairs.
The Formula
a/sin(A) = b/sin(B) = c/sin(C)
Used for: Finding unknown sides or angles in any triangle
a, b, cLengths of sides
A, B, CAngles opposite to sides
🎯 When to Use the Sine Rule
Best for triangles where you know corresponding angle-side pairs:
📊
Works for ANY triangleNot just right triangles!
🔄
Sides match opposite anglesSide a is across from angle A
📐
Need 2 known pairsUse when you know angle + opposite side
📝 Worked Examples
Find a Side
Given:
A = 30°, a = 5, B = 60°. Find b.1
Set up ratio
a/sin(A) = b/sin(B)2
Substitute
5/sin(30°) = b/sin(60°)3
Solve
b = 5 × sin(60°)/sin(30°)4
Calculate
b = 5 × 0.866/0.5 = 8.66Answer: b ≈ 8.66
Find an Angle
Given:
a = 8, A = 40°, b = 10. Find B.1
Set up ratio
sin(A)/a = sin(B)/b2
Solve for sin(B)
sin(B) = b × sin(A)/a3
Calculate
sin(B) = 10 × sin(40°)/84
Inverse sine
B = sin⁻¹(0.804) ≈ 53.5°Answer: B ≈ 53.5°
💡 Pro Tips
Ambiguous case
When finding angles, sin⁻¹ may give two possible answers (acute or obtuse)
Use degrees or radians
Just be consistent! Check your calculator mode.
When to use
Use sine rule when you have angle-side pairs; cosine rule otherwise