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Sine Rule (Law of Sines)
Trigonometry โข Grades 9, 10, 11
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 side lengths or angle measures in any non-right oblique triangle
a, b, cLengths of triangle sides
A, B, CAngles opposite to sides a, b, c
โกLive Interactive Quick Solver
Instant CalculationSide b (opposite B):5.20 units
b = (3 ร sin(60ยฐ)) / sin(30ยฐ) = 5.20โข Angle C = 180ยฐ - 30ยฐ - 60ยฐ = 90ยฐ
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
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๐ฏ When to Use the Sine Rule
Best for triangles where you know corresponding angle-side pairs:
๐
Works for ANY triangleNot just right triangles!
๐
Sides match opposite anglesSide a is across from angle A
๐
Need 2 known pairsUse when you know angle + opposite side
๐ Worked Examples
Find a Side
Given:
A = 30ยฐ, a = 5, B = 60ยฐ. Find b.1
Set up ratio
a/sin(A) = b/sin(B)2
Substitute
5/sin(30ยฐ) = b/sin(60ยฐ)3
Solve
b = 5 ร sin(60ยฐ)/sin(30ยฐ)4
Calculate
b = 5 ร 0.866/0.5 = 8.66Answer: b โ 8.66
Find an Angle
Given:
a = 8, A = 40ยฐ, b = 10. Find B.1
Set up ratio
sin(A)/a = sin(B)/b2
Solve for sin(B)
sin(B) = b ร sin(A)/a3
Calculate
sin(B) = 10 ร sin(40ยฐ)/84
Inverse sine
B = sinโปยน(0.804) โ 53.5ยฐAnswer: B โ 53.5ยฐ
๐ก Pro Tips
Ambiguous case
When finding angles, sinโปยน 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
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