Section 9.3 – The Law of Cosines In this section we will use the Law of Cosines to solve Case 3 (SAS) and Case 4 (SSS) triangles.
It is really unnecessary to write out all three of these equations, because they all display the same concept: "The square of one side of a triangle equals the sum of the squares of the other two sides minus 2 times their product times the cosine of the angle between them."
The Law of Cosines, as shown above, makes it easy to find a missing side (remember that the lower-case letters are side lengths and the CAPTIAL letters are angle measurements). But often we need to use the Law of Cosines to find angle measurements. So let's solve these equations for A, B, and C. We'll start by using the first equation in the box above. 2
2
subtract a and b -2ab c 2 a 2 b 2 2ab cos C c 2 a 2 b 2 2ab cos C
Now, factoring out a -1 in the numerator gives:
1 c 2 a 2 b 2 2ab
c 2 a 2 b2 cos C . 2ab
cos C .
c 2 a 2 b 2 a 2 b2 c 2 Cancelling the negative signs gives: cos C , and rewriting the numerator gives: cos C . 2ab 2ab a 2 b2 c 2 . 2ab
So, C cos 1
Applying the same concept to the other two angles, we get the following formulas:
2 2 2 2 2 2 b2 c 2 a2 1 a c b 1 a b c A cos 1 , B cos , and C cos . 2 bc 2 ac 2 ab
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SOLVE CASE 3 (SAS) TRIANGLES Example: Solve each triangle. a) a = 2, c = 1, B = 10 We will start by finding the length of side b.
b 2 a 2 c 2 2ac cos B b 2 2 2 1 2 2 2 1 cos10 b 2 1.06077 b 1.03 Now we can use the Law of Sines (from Section 9.2) OR the Law of Cosines to find the two remaining angles, A and C. If we use the Law of Sines, we have the following:
sin A sin B sin A sin10 2sin10 sin A sin A 0.337 A 19.7 or A 160.3 a b 2 1.03 1.03 sin C sin B sinC sin10 1sin10 sinC sin C 0.1686 C 9.7 or C 170.3 c b 1 1.03 1.03 Remember that you have to find both the Q1 and Q2 angles if you use the Law of Sines. Since this gives us two answers, we have to figure out which one is correct. We know for sure that B = 10 because that is given in the problem. Then C can NOT equal 170.3 because B C 10 170.3 180.3 180 . So C must equal 9.7. If B = 10 and C = 9.7, then A must equal 160.3 because A B C 180 160.3 10 9.7 180 . If we had found C first, then we actually wouldn’t need to use the Law of Sines to find A. We could just use the fact that
A B C 180 , so A 180 10 9.7 160.3 . If we decided to find the measurements of A and C using the Law of Cosines instead of the Law of Sines, then we wouldn’t have to "figure out" which angle to use. Because cosine is positive in quadrants 1 and 4 and negative in quadrants 2 and 3, and because all of the angles of a triangle must add up to 180, then it is not possible for an angle to come from quadrants 3 or 4 (because all of the angles in those quadrants are greater than _____________) . Thus, cos-1(positive #) must come from Q1 and cos-1(negative #) must come from Q2. We will never find two answers if we use the Law of Cosines the way we do with the Law of Sines. Let's demonstrate this now, using the equations from the bottom of page 1: 2 2 2 b2 c 2 a 2 1 1.03 1 2 A cos 1 cos 2bc 2 1.03 1
1 1.9391 cos 160.3 2.06
2 2 2 a 2 b2 c 2 4.0609 1 2 1.03 1 C cos 1 cos cos 1 9.7 2 2 1.03 2ab 4.12
It is entirely your choice whether you want to use the Law of Sines or the Law of Cosines in these problems, but using the Law of Cosines ends up being less work since you don't have to figure out which answer to use like you do with the Law of Sines. Page | 2
Example (continued): Solve each triangle. b) a = 6, b = 4, C = 60 Start by finding the length of side c.
c 2 a 2 b 2 2ab cos C c 2
Now use the Law of Sines or the Law of Cosines to find the measurements for angles A and B.
c) b = 4, c = 1, A = 120 Start by finding the length of side a.
a 2 b 2 c 2 2bc cos A a 2
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SOLVE CASE 4 (SSS) TRIANGLES Example: Solve each triangle. a) a = 4, b = 5, c = 3 Since we are going to be solving for angles in this problem, use the angle formulas at the bottom of page 1:
b2 c 2 a2 A cos 1 2bc
2 2 2 1 5 3 4 1 18 cos cos 53.1 30 2 5 3
2 2 2 a2 c 2 b2 1 4 3 5 1 B cos 1 cos cos 0 90 or ________ (but the latter isn't possible – 2 ac 2 4 3 why?_______________________________________________________________________________________) At this point, we could finish the problem by saying that C = 180 A B = 180 53.1 90 = 36.9. Let's see if this is what we get using the Law of Cosines formula: 2 2 2 a 2 b2 c 2 1 4 5 3 C cos 1 cos 2 4 5 2ab
1 32 cos 36.9 40
So our final answer is: A 53.1, B 90, C 36.9 This actually is a ____________ __ triangle!
b) a = 9, b = 7, c = 10
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Example (continued): Solve each triangle. c) a = 3, b = 3, c = 2
SOLVING APPLICATION PROBLEMS USING THE LAW OF COSINES Example: #34) An airplane flies due north from Ft. Myers to Sarasota, a distance of 150 miles, and then turns through an angle of 50 and flies to Orlando, a distance of 100 miles. (a) How far is it directly from Ft. Myers to Orlando? (b) What bearing should the pilot use to fly directly from Ft. Myers to Orlando?
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Example: #36) In attempting to fly from Chicago to Louisville, a distance of 330 miles, a pilot inadvertently took a course that was 10 in error. (a) If the aircraft maintains an average speed of 220 miles per hour and if the error in direction is discovered after 15 minutes, through what angle should the pilot turn to head toward Louisville? (b) What new average speed should the pilot maintain so that the total time of the trip is 90 minutes?
Example: #38) According to Little League baseball official regulations, the diamond is a square 60 feet on a side. The pitching rubber is located 46 feet from home plate on a line joining home plate and second base. (a) How far is it from the pitching rubber to first base? (b) How far is it from the pitching rubber to second base? (c) If a pitcher faces home plate, through what angle does he need to turn to face first base?
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