Mathematics
The Law of Sines
The Law of Sines relates the sides of a triangle to the sines of their opposite angles: a/sin(A) = b/sin(B) = c/sin(C). Use it when you know an angle and its opposite side, plus one more piece.
How to Approach The Law of Sines
Check the case
You need either AAS (angle, angle, side) or ASA (angle, side, angle), or SSA (side, side, angle — the ambiguous case). The Law of Sines doesn't work for SAS or SSS.
Set up the proportion
Write a/sin(A) = b/sin(B). Substitute the known values. Solve for the unknown — either a side using cross-multiplication, or an angle using inverse sine.
Watch for the ambiguous case
In SSA, the inverse sine might have two valid angles (θ and 180°-θ). Check both against the triangle inequality to see if both are geometrically possible.
Frequently Asked Questions
When does the Law of Sines fail?+
When you have SAS (two sides and the included angle) or SSS (three sides). Use the Law of Cosines for those.
What's the ambiguous case?+
When you're given two sides and a non-included angle (SSA), the triangle might have zero, one, or two valid solutions. Always check the supplementary angle too.
Is it the same on the SAT?+
Law of Sines isn't directly tested on SAT but appears on the ACT, AP Pre-Calculus, AP Calculus AB context problems, and the IB syllabus.
Related Topics
More step-by-step guides in Mathematics and adjacent subjects.
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