When solving to for the sides of a right triangle, we use the Pythagorean Theorem, where a2 +b2 = c2. Once we know the sides, we can use trigonometric functions sine, cosine and tangent to find the measures of the other two angles. An oblique triangle is one that is not right. But how does one solve for the sides and angles of an oblique triangle? The Law of Sines is used if a side and two angles are known, two angles and the sides between them are known, or two sides and an angle opposite one side is known. The formula is very easy, sinA/a = sinB/b = sinC/c. In the case of two sides and included angle or all three sides given, the Law of Cosines must be used. What is the Law of Cosines?
The Law of Cosines may look confusing at first, but is quite simple to use and memorize. If A, B, and C are the measures of the angles of a triangle, and a, b and c are the lengths of the sides opposite of those angles, then a2 = b2 + c2- 2bccosA, b2 = a2 + c2 – 2accosB and c2 = a2 + b2 – 2abcosC. Notice the side we are solving for and the angle we take the cosine of. It’s the angle opposite the side. The other two sides are the ones we square on the other side of the equation and multiplied together by 2.
To solve a SAS triangle, first use the Law of Cosines to find the side opposite the angle given. Then use the Law of Sines to find the angle opposite the shorter of the two given sides. Then find the third angle by subtracting the sum of the measures of the two angles from 180.
Example: Solve the triangle with A = 50 degrees, b = 15, c = 20.
Using Law of Cosines we get a2 = (15)2 + (20)2 – 2(15)(20)cos(50)
= 225 + 400 – 600cos(50)
= 625 – 385.67
a = 15.47
Find angle B using Law of Sines.
SinB/15 = Sin(50)/15.47
SinB = 0.7
B = 48 degrees
Angle C is 180 – (50 + 48) = 180 – 98 = 82.
To solve a SSS triangle, use the Law of Cosines to find the angle opposite the longest side. Then use the Law of Sines to find either of the other remaining angles. Then subtract the sum of the other two angles from 180.
Example: Solve the triangle if a = 6, b = 10, c = 12.
Use the Law of Cosines to find angle C.
c2 = a2 + b2 – 2abcosC
(12)2= (6)2 + (10)2 – 2(6)(10)cosC
144 = 36 + 100 – 120cosC
144 = 136 – 120cosC
8 = -120cosC
-0.066 = cosC
C = 93.8
Use Law of Sines to get angle A
SinA/6 = Sin(93.8)/12
SinA/6 = 0.083
SinA = 0.499
A = 29.9 degrees
B = 180 – (93.8 + 29.9) = 180 – 123.7 = 56.3 degrees
Notice how the Law of Cosines always solves for a part of the triangle which enables you to use the Law of Sines. I’ve been using example such as the one’s above during my 14 years of tutoring trigonometry. They should help any student who is having difficulty using the Law of Cosines.