The law of sines says that in any triangle, each side divided by the sine of its opposite angle gives the same number: asinA = bsinB = csinC. Use it to solve triangles that have no right angle when you know two angles and a side, or two sides and an angle opposite one of them.
Each fraction pairs an angle with the side across from it. Know one full pair and one more piece, and you can find the rest.
Formal definition and notation
In a triangle with angles A, B, C and opposite sidesa, b, c, sinAa = sinBb = sinCc, equivalently asinA = bsinB = csinC. It solves the AAS and ASA cases uniquely. In the SSA case (two sides and a non-included angle) the data may give no triangle, one triangle or two triangles, called the ambiguous case, because sinB = k has two solutions between 0° and 180° when 0 < k < 1.
Sine is a single ratio of one angle. The law of sines is an equation that links the sines of all three angles of a triangle with its sides.
Quick check: 3 questions
Question 1 of 3
Which proportion is the law of sines?
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Answer: A. Each side goes with the sine of the angle across from it. The other three pair sides with the wrong angles or multiply instead of divide.
Question 2 of 3
In triangle ABC, A = 30°, a = 6 and B = 90°. What is b?
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Answer: C.bsin 90° = 6sin 30° = 60.5 = 12. 3 multiplies by 0.5 instead of dividing, and 6√3 is the third side c.
Question 3 of 3
Which set of facts can be solved with the law of sines right away?
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Answer: C. With two angles you know the third, and any side then pairs with its opposite angle. Three sides and SAS need the law of cosines, and three angles fix only the shape, not the size.
Law of Sines: frequently asked questions
When do you use the law of sines?
Use it in a triangle without a right angle when you know an angle and the side across from it, plus one more side or angle: the AAS, ASA and SSA cases. If you know two sides and the angle between them, or all three sides, use the law of cosines instead.
What is the ambiguous case of the law of sines?
It is the SSA case: two sides and an angle that is not between them. Solving gives sinB, and two angles between 0° and 180° share that sine, an acute one and 180° minus it. Depending on the numbers, zero, one or two triangles fit, so always test the second angle.
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