Math 150 – Fall 2015
Section 8G
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Section 8G – Law of Sines and Law of Cosines
The Law of Sines
Theorem. For any triangle , the ratio of the sine of an angle to the length of the side
opposite the angle does not depend on which vertex of the triangle we use. That is
sin B
sin C
sin A
=
=
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 Law of Cosines
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
Note. This is a generalization of Pythagorean’s Theorem. If the angle C is π2 or 90◦ ,
then then −2ab cos C = −2ab cos π2 = 0, and the Law of Cosines is exactly the same as
Pythagorean’s Theorem: c2 = a2 + b2 .
Math 150 – Fall 2015
Section 8G
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Solving Triangles
Give some information about a triangle, we would like to solve for the lengths of all the
sides and the measure of all the angles.
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 – Fall 2015
Example 2. Given a = 10, A =
Section 8G
π
7,
and B =
2π
7 ,
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solve the triangle.
√
Example 3. Given A = 30◦ , a = 5, and b = 5 2, solve the triangle.
Math 150 – Fall 2015
Section 8G
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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
between them is sin−1 25 , what is the length of the third side of the triangle?