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1 8.2 The Law of Sines Section 8.2 Notes Page 1 The law of sines is used to solve for missing sides or angles of triangles when we have the following three cases: S ngle Side ngle S Side ngle ngle SS Side Side ngle Law of Sines sin sin sin = = b a b c c a Usually you will only use two parts of the above formula, but all three ratios are equal. EXMPLE: Solve the triangle: 60 0 The first thing we can find is the measurement of angle by subtracting the given angles from 180 degrees: m = = 80. Next we can find side, which I will label as b. We will use the law of sines for this. You always need to start with a known side and a side opposite the known side. I will use the 0 degrees and the. This will be one side of the equation. The other side is for the side you want to find. Since we want to find side we sin 60 need to use the angle opposite of this side, so we will use the 60 degrees. The equation is: =. b sin 60 ross multiplying will give us b = sin 60. Solving for b we get: b = Now we want to find side, which I will call c. Once again we will start with a known angle and side sin 80 opposite this angle. The equation is: =. ross multiplying will give us c = sin 80. c sin 80 Solving for c we get c = Now the triangle is solved.

2 EXMPLE: Solve the triangle: Section 8.2 Notes Page The first thing we can find is the measurement of angle by subtracting the given angles from 180 degrees: m = = 130. Next we can find side, which I will label as b. We will use the law of sines for this. I will use the 130 sin130 sin15 degrees and the 5 as my known angle and side. The equation is: =. ross multiplying will 5 b 5sin15 give us b sin 130 = 5sin15. Solving for b we get: b = sin130 Now we want to find side, which I will call a. Once again we will start with a known angle and side sin130 sin 35 opposite this angle. The equation is: =. ross multiplying will give us a sin 130 = 5sin a 5sin 35 Solving for c we get a = Now the triangle is solved. sin130 EXMPLE: Solve the triangle: We first want to find one of the missing angles. The only one I can solve for is angle since I have a side sin opposite that is given: =. Now cross multiply: 2 = 3sin. Solving for sin we get: sin =. So sin = Now we will take the inverse sine of both sides to get m = Now we can find angle : m = = sin11.63 Now that we know angle we can find side. I will call this c: =. ross multiplying 3 c 3sin11.63 gives us c = 3sin Solving for c we get c =. 2.

3 EXMPLE: Solve the triangle: Section 8.2 Notes Page 3 50 I want to solve for angle first since there is a side opposite this angle given. We can use the equation: sin 50 sin =. ross multiplying will give us: 2 sin 50 = sin. So sin = we try and take the 1 2 inverse of this in our calculator we will get an error. This is because the domain of the inverse sine function must be between -1 and 1. There is no solution for this problem. That means it is impossible to draw this triangle. The drawing above is not to scale. we did draw it to scale it would look something like this: Notice that the side with a 1 is not long enough, so we can t complete the triangle. gain the answer is no solution. EXMPLE: Solve the triangle: 25 I showed on the board that the triangle above could be drawn two different ways. The first way is the above drawing. we take side and swing it to the left, we will get a second triangle: 25 sin 25 sin You will actually get two solutions algebraically. Let s try and solve for angle again: =. 1 2 ross multiplying gives us: 2 sin 25 = sin. This will give us sin = The inverse sine will give us: = Now the calculator just gives us this one angle, which would correspond to our first drawing. Remember that sine is positive in the first and second quadrant, so if 57.7 is a reference angle, then we find another angle in the second quadrant by subtracting this from 180 degrees: m = = So we are going to get two solutions. We will find two different angle s and two different side s. Go to next page to see both solutions worked out.

4 Triangle 1: Section 8.2 Notes Page m = 57.7 then we can find angle : m = = 97.3 sin 25 sin 97.3 We can use this to find side, which I will call b: =. ross multiplying will give us: 1 b sin 97.3 b sin 25 = sin Solving for b we get: b = sin 25 Triangle 2: m = then we can find angle : m = = 32.7 sin 25 sin 32.7 We can use this to find side, which I will call b: =. ross multiplying will give us: 1 b sin 32.7 b sin 25 = sin Solving for b we get: b = sin 25 EXMPLE: Solve the triangle: 0 3 The only angle we can solve for here is angle since we have a side opposite angle that is given: sin 3 =. ross multiplying will give us 3 = sin. Then we have: sin =. This 3 will give us: sin = When we find the inverse sine our calculator gives us m = Now we need to see if there will be another solution. Once again sine is positive in the second quadrant, so we can find a second solution by subtracting our answer from 180 degrees: m = = Notice that we already have an angle in the triangle that is 0 degrees, so it is impossible to also have an angle of degrees because then the sum of the angles would be more than 180 degrees. Therefore we can ignore this second solution. We know for sure that m = Now we can find angle by subtracting from 180 degrees: m = = Finally we can find side, which I will label as a: sin =. ross multiplying gives us: a = sin Solving for a will give us: sin a = 5.8. More on next page

5 EXMPLE: Solve the triangle: 5 Section 8.2 Notes Page 5 0 sin We want to find angle since we have a side opposite angle that is given to us: =. ross 5 5 multiplying will give us: 5 = sin. Then we have: sin =. So sin = The inverse sine gives us m = Our other solution for angle is: m = = we add this to the 0 degree angle already in the triangle we will get an angle less than 180 degrees, so this tells us there will definitely by two solutions to this triangle. The first solution is the drawing shown above. we take side and swing it to the left, we will get a second triangle: 5 0 Now we will solve both triangle separately: Triangle 1: m = 53.6 then we can find angle : m = = 86.5 sin 86.5 We can use this to find side, which I will label a: =. ross multiplying will give us: sin 86.5 a = sin Solving for b we get: a = Triangle 2: m = then we can find angle : m = = 13.6 sin13.6 We can use this to find side, which I will label a: =. ross multiplying will give us: sin13.6 a = sin13.6. Solving for a we get: a = 1. 5.

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