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Trigonometry > Law of Cosines
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QuickNotes™ 
Law of Cosines
Main formulas:
- The Law of Cosines:
This holds in any triangle - not just right triangles! This is a generalization of the Pythagorean Theorem. (If C is a right angle, then cosC = 0, so we get c2 = a2 + b2.)c2 = a2 + b2 − 2abcosC - Use the Law of Cosines for triangles described in the following ways:
- SAS always has a unique solution (assuming the angle is less than 180° ).
- SSS always has a unique solution (assuming each side is less than the sum of the other two).
- (Use Law of Sines for ASA, SAA, and SSA.)
- Heron's Formula:
where s = [1/2](a+b+c) is the semiperimeter of the triangle.Area =
√
s(s− a)(s− b)(s− b),
Example 1:
In triangle ABC, side a has length 3, side b has length 4, and angle C measures 60° . Determine how many triangles satisfy these conditions and solve the triangle(s) completely.Example 2:
The side lengths of triangle ABC are 5, 7, and 10. Determine how many triangles satisfy these conditions and solve the triangle(s) completely.Example 3:
Find the area of a triangle whose side lengths are 5, 7, and 10.Example 4:
The side lengths of triangle ABC are 16, 30, and 34. Determine how many triangles satisfy these conditions and solve the triangle(s) completely.Example 5:
A triangle has two sides of length 8 and 16 with an included angle of 45° . Find the length of the third side and the area of the triangle.Discussion
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