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Trigonometry > Law of Sines
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QuickNotes™ 
Law of Sines
Main formulas:
- The Law of Sines:
This holds in any triangle - not just right triangles!sinA a= sinB b= sinC c - Use the Law of Sines for triangles described in the following ways:
- ASA always has a unique solution (assuming the angles sum to less than 180° ).
- SAA always has a unique solution (assuming the angles sum to less than 180° ).
- SSA might have no solutions, one solution, or two solutions.
- AAA always has infinitely many solutions (assuming the angles sum to 180° ), but the triangles are all similar.
- (Use the Law of Cosines for SAS and SSS.)
Example 1:
In triangle ABC, angle A measures 50° , angle B measures 60° , and side c has length 4. Determine how many triangles satisfy these conditions and solve the triangle(s) completely.Example 2:
In triangle ABC, side a has length 3, side b has length 4, and angle A measures 40° . Determine how many triangles satisfy these conditions and solve the triangle(s) completely.Example 3:
In triangle ABC, side a has length 7, side b has length 12, and angle A measures 45° . Determine how many triangles satisfy these conditions and solve the triangle(s) completely.Example 4:
In triangle ABC, angle A measures 40° , angle B measures 110° , and side a has length 7. Determine how many triangles satisfy these conditions and solve the triangle(s) completely.Example 5:
In triangle ABC, side a has length 10, side b has length 8, and angle B measures 20° . Determine how many triangles satisfy these conditions and solve the triangle(s) completely.Discussion
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