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# The law of sines

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Intro to Law of Sines

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### The law of sines

1. 1. Cassandra Currier Internet In Education EME 6053
2. 2. The Law of Sines
3. 3. What if the triangle we want to solve is NOT a right triangle? In the next two sections we’ll develop ways of solving these triangles if we know at least one side and two other pieces of info (sides or angles or one of each). B c A a C b We’ll label these triangles with sides a, b and c and use their Uppercase alphabet counterparts for the angles opposite those sides as shown above.
4. 4. B c a h A C b Draw a perpendicular line and call the length h. We do this so that we have a right triangle which we already know how to work with. Let’s write some trig functions we know from the right triangles formed. sin A h c Solve these for h c sin A h sin C Since these both = h we can substitute a sin C divide both sides by ac c sin A a sin C ac ac sin A a sin C c h a h
5. 5. This process can be repeated dropping a perpendicular from a different vertex of the triangle. What we get when we combine these is: THE LAW OF SINES sin A a sin B b sin C c Use these to find missing angles THE LAW OF SINES a sin A b sin B c sin C Use these to find missing sides What this says is that you can set up the ratio of the sine of any angle in a triangle and the side opposite it and it will equal the ratio of the sine of any other angle and the side opposite it. If you know three of these pieces of information, you can then solve for the fourth.
6. 6. Solve a triangle where B = 55 , C = 82 and c = 9 Draw a picture (just draw and 9 c label a triangle. Don't worry about having lengths and This is SAA angles look right size) 43 A Do we know an angle and side opposite it? If so the Law of 9 b Sines will help us determine the other sides. sin 82 sin 55 How can you find ? 9 sin 55 sin 82 b Hint: The sum of all the angles in a triangle is 180 . A 180 55 82 43 a sin 43 9 sin 82 a 55 B 6.20 a C 82 7.44 b b sin 82 9 sin 55 7.44 How can you find a? (Remember it is NOT a right triangle so Pythagorean theorem will not work). 9 sin 43 sin 82 6.20 You can use the Law of Sines again.
7. 7. There are three possible configurations that will enable us to use the Law of Sines. They are shown below. You don’t have an angle and side opposite it here but can easily find the angle opposite the side you know since the sum of the angles in a triangle must be 180 . ASA You may have an angle, a side and then another angle SSA You may have two sides and then an angle not between them. SAA You may have a side and then an angle and then another angle What this means is that you need to already know an angle and a side opposite it (and one other side or angle) to use the Law of Sines.
8. 8. Let's look at a triangle where you have SSA. b a A It could be that you can't get the sides to join with the given info so these would be "no solution". b It could be that there is one triangle that could be formed and you could solve the triangle. A b b A a a A a, b and A are the same in both of these triangles. It could be that since side c and C are not given that there are two ways to draw the triangle and therefore 2 different solutions to the triangle a
9. 9. C h b sin A b h A c a B •If A is acute and a < h, then there are no solutions. •If A is acute and a = h, then there is one solution. •If A is acute and a > b, then there is one solution. •If A is acute and h < a< b, then there are two solutions. •If A is obtuse and a ≤ b, then there are no solutions. •If A is obtuse and a > b, then there is one solution.
10. 10. You can just check to see if there are two triangles whenever you have the SSA case. The "no solution" case will be obvious when computing as we will see. Solve a triangle where B = 95 , b = 4 and c = 5 C 4 We have SSA. We know an angle and a side opposite it so we'll use the Law of Sines. a A 95 5 sin C 1.245 C sin 1 1.245 sin 95 4 sin C 5 5sin 95 4 sin C We have the answer to sine and want to know the angle so we can use inverse sine. What happens when you put this in your calculator? Remember the domain of the inverse sine function is numbers from -1 to 1 since the sine values range from -1 to 1. What this means is there is no solution. (You can't build a triangle like this).
11. 11. Solve a triangle where B = 35 , b = 6 and c = 8 49.9 C The smallest angle should have the smallest side opposite it and the largest angle should have the largest side opposite it. 6 a 10.42 sin 35 6 95.1 A 35 8 sin C 0.765 C sin 1 Now how can you find a? sin C 8 0.765 Knowing B and C, can you find A? sin 35 6 We have SSA again so we know it could be the weird one of no solution, one solution or two solutions. C 8sin 35 6 sin C 49.9 180 35 49.9 95.1 sin 95.1 a a 6 sin 95.1 10.42 sin 35 Since this was an SSA triangle we need to check to see if there are two solutions. Remember your calculator only gives you one answer on the unit circle that has the sine value of 0.765. You need to figure out where the other one is and see if you can make a triangle with it.
12. 12. Looking at the same problem: Solve a triangle where B = 35 , b = 6 and c = 8 130.1 C Let's check to see if there is another triangle possible. We got 49.9 from the calculator. Draw a picture and see if there is another angle whose sine is 0.765. 6 2.69 a 180 - 49.9 = 130.1 14.9 A 35 49.9 So there IS another triangle. (remember our picture is not drawn to scale). 8 sin C 0.765 Knowing B and C, can you find A? 180 Now how can you find a? 6 sin 35 35 130.1 14.9 a sin 14.9 a 6 sin 14.9 sin 35 2.69
13. 13. Solve a triangle where B = 42 , b = 22 and c = 12 21.4 C sin 42 22 22 sin C 12 sin C 0.365 1 C sin 0.365 21.4 29.40 a 116.6 A 42 Since this is SSA we need to check the other possible sine value for possibility of a second triangle solution. 12 Knowing B and C, can you find A? sin 42 22 a sin 116.6 a 180 42 21.4 116.6 22 sin 116.6 sin 42 29.40
14. 14. Solve a triangle where B = 42 , b = 22 and c = 12 158.6 C Not possible to build another triangle with these stipulations. 22 a A 42 180 - 21.4 = 158.6 21.4 12 Knowing B and C, can you find A? 180 42 158.6 20.6 This negative number tells us that there is no second triangle so this is the one triangle solution.