Math 150 – c Lynch 1 of 4

Section 9.5 – Law of and Cosines

The Law of Sines Theorem. For any , the ratio of the of an to the length of the side opposite the angle does not depend on which vertex of the triangle we use. That is sin A sin B sin C = = a b c using the notation from the triangle below.

Example 1. A triangle has two sides of length 3 inches and 5 inches with the angle opposite the side of length 5 inches equal to 25◦. Determine the length of the third side.

The Theorem. If a, b, and c are the lengths of the sides of any triangle and C is the angle opposite the side of length c (equivalently, C is the angle included by the sides with lengths a and b), we have c2 = a2 + b2 − 2ab cos C

π ◦ Note. This is a generalization of Pythagorean’s Theorem. If the angle C is 2 or 90 , π then then −2ab cos C = −2ab cos 2 = 0, and the Law of Cosines is exactly the same as Pythagorean’s Theorem: c2 = a2 + b2. Math 150 – c Lynch 9.5–Law of Sines and Cosines 2 of 4

Solving Give some information about a triangle, we would like to solve for the lengths of all the sides and the measure of all the . Theorem. A triangle is uniquely determined if we know one of the following:

• SSS – All three sides. • SAS – two sides and the angle included between them. • ASA – Two angles and the side between the angles

• SAA – Two angles and the side opposite one of the angles. Note that SSA – two sides and an angle that is opposite one of the sides (so it is not the angle included between them) – does NOT uniquely determine a triangle.

How to solve a triangle: A triangle has six parts: three angles and three sides. Solving a triangle means finding all six parts. Use the following instructions to solve a triangle based on the given information:

• ASA, SAA: find the third angle, use Law of Sines for the remaining sides. • SSA: use Law of Sines to get the angle opposite the given side (there may be two possible angles). For all cases, find the third angle and use Law of Sines for the remaining side. • SAS: Use the Law of Cosines to find the third side. Use the Law of Sines to solve the smaller of the two angles first (Hint: we know it’s less than 90◦). • SSS: Use the Law of Cosines to find an angle. Then use the Law of Sines or Cosines for the remaining angles. Note. • All three angles must add up to 180◦ or π radians.

• The largest angle is opposite the largest side, and the smallest angle is opposite the smallest side. • The largest angle might or might not be obtuse. The smaller two angles must be acute (less than 90◦ or π). Math 150 – c Lynch 9.5–Law of Sines and Cosines 3 of 4

π 2π Example 2. Given a = 10, A = 7 , and B = 7 , solve the triangle.

√ Example 3. Given A = 30◦, a = 5, and b = 5 2, solve the triangle. Math 150 – c Lynch 9.5–Law of Sines and Cosines 4 of 4

Example 4. Find the three angles of the triangle with sides of length 2.5, 4, and 5.3.

Example 5. Given a = 1, b = 2, and A = 45◦, solve the triangle.

Example 6. Two sides of a triangle are of lengths 3 meters and 2 meters. If the angle −1 2 between them is sin 5 , what is the length of the third side of the triangle?