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### Course: Geometry (all content)>Unit 13

Lesson 9: The law of sines

# Solving for an angle with the law of sines

Sal is given a triangle with two side lengths and one angle measure, and he finds all the missing angle measures using the law of sines. Created by Sal Khan.

## Want to join the conversation?

• At , why do we take the inverse sine instead of dividing both sides by sine to get rid of the sine of the right side?
(42 votes)
• You can't divide the sines of two different angles. On scene , our angles are theta and 40 degrees. Although dividing by sin(theta) would remove the sine from the right side, you would only be left dividing the sine of 40 degrees and the sine of theta on the left side. However, you can use inverse sine and remove the sines that way, because the inverse sine of the sine of 40 degrees is 40 degrees and the inverse sine of the sine of theta is theta.
(71 votes)
• I didn't understand the inverse sin step. Can anyone please explain to me what it is?
(24 votes)
• The inverse sin gets him to theta by itself
(4 votes)
• Is there a Law Of Tangents?
(13 votes)
• Yes there is. The Law of Tangents is a statement involving the tangents of two angles in a triangle and the lengths of opposite sides.

The Law of Tangents state:
a-b/a+b = tan[1/2(A-B)]/tan[1/2(A+B)]
(19 votes)
• If you search the law of sines on the internet, it'll mostly give you A/sin(a) = B/sin(b) = C/sin(c). But, to find a missing angle, it's best to use sin(a)/A = sin(b)/B = sin(c)/C. Is there a way to use
A/sin(a) = B/sin(b) = C/sin(c) to find a missing angle?
(7 votes)
• If you have A/sin(a) = B/sin(b), you can just solve for the angle using Algebra.
A = B * sin(a) / sin(b)
A * sin(b) / B = sin(a)
a = arcsin(A * sin(b) / B)
(18 votes)
• Hi, I have a question. I have an hp35s calculator.
1. I did the following question on law of sines-
sin14°13'36"/803.94=sinX/1879.28
I get the answer 35°03'48". Which is correct, but the answer could also be 144°56'12". Why is that? I only ask because a question I was doing required me to know that.
2.Your example of the law of sines.
sin40/30=sinx/40
I got an answer of 58°59'13"
I think I did this as precise as I could because the question I did above came out correctly and is the same format. I'm sure you weren't getting into that precise of an answer, but in my situation, I must have it that precise for a test. So, my question for this one- would my answer be correct in terms of pricision?

Thank you,

Jerry
(9 votes)
• 145 degrees is just the same as 35 degrees but pointing in the opposite direction. The vertical height of the triangle is the same.
(6 votes)
• What is the difference between degree mode and radian mode? Thanks
(4 votes)
• You said mode in your question, so I suppose you are refering to calculators. Degree mode will set up your calculator to work with degrees.So in any operation that involves an angle amplitude or trig ratio, your results will be in degrees. In radian mode, your calculator will set up to work with radians, that means, a different unit of amplitude. Always remember to switch modes acordingly to the unit you are using, else you will get some pretty weird answers.

Hope this helps.
(10 votes)
• Why Sal doesn't consider that theta angle might be 180 - 58.9869695348 ? There's the second solution: 18.9869695348
(5 votes)
• You are correct. Since the triangle is in the form SSA there is ambiguity as sin(180-theta) = sin(theta).

I have notified Khan Academy of this mistake. Hopefully it will get rectified in the future.
(5 votes)
• Does it matter that much at whether a capital "A" is used or a lowercase "a" is used in the low of sines.

Example; sinA/a= sinB/b=sinC/c if the triangle angle are the capital 'A"'s
(4 votes)
• Nope- the letters don't matter- they only stand for words. as long as the values are being divided correctly (AKA, the angle of sine are divided by the corresponding side length), you could call one angle "Marshmallow" and another angle "Cheeseburger" and it wouldn't make one smidgeon of a difference.
(6 votes)
• I'm not sure if this is just another way to do it, or it worked this once by luck, but about into the video I paused it because something clicked when he was talking about the constant ratio between all sides and angles in a triangle, and I immediately went to google calculator, found the constant ratio, multiplied it by 40(the measure of the side, not the angle) and then did the inverse sine of that, and I unpaused the video to see if I got it correct(I did). Will this always work?
(6 votes)
• 👍Of course, it is right! you can test this in the practice btw! This is pure genius since you put it constant ratio. Great Job!

Hope this helped you understand and get encouraged!

#YouKhanLearnAnything.💪
(3 votes)
• I was doing the problems, I got an answer, but it was incorrect. I checked what the solution was and it said that sin(theta) = sin(180 - ANSWER) Can anyone explain how this works? It was not explained in the video.
(3 votes)
• Sometimes when using the law of sines, you can have a situation where there could be two possible triangles. We call this the ambiguous case, and I highly recommend you search that phrase up. Since from 0-180 degrees (the angles that you work with for triangles), sine has the same value twice, there is some uncertainty as to which angle to choose from the two that you could get. In your case, substituting theta in for the angle may have made the sum of angle measures more than 180. You would then need to take the other value that sine provides, and use that, which may/may not work (there's a possibility of no triangle existing that can fulfill the problem's conditions).
(6 votes)

## Video transcript

Voiceover:Say you're out flying kites with a friend and right at this moment you're 40 meters away from your friend and you know that the length of the kite's string is 30 meters, and you measure the angle between the kite and the ground where you're standing and you see that it's a 40 degree angle. What you're curious about is whether you can use your powers of trigonometry to figure out the angle between the string and the ground. I encourage you to pause the video now and figure out if you can do that using just the information that you have. Whenever I see, I guess, a non right triangle where I'm trying to figure out some lengths of sides or some lengths of angles, I immediately think maybe the Law of Cosine might be useful or the Law of Sines might be useful. So, let's think about which one could be useful in this case. Law of Cosines, and I'll just rewrite them here. The Law of Cosine is c squared is equal to a squared plus b squared minus 2ab cosine of theta. So what it's doing is it's relating 3 sides of a triangle. So a, b, c to an angle. So, for example, if I do 2 sides and the angle in between them, I can figure out the third side. Or if I know all 3 sides, then I can figure out this angle. But that's not the situation that we have over here. We're trying to figure out this question mark and we don't know 3 of the sides. We're trying to figure out an angle but we don't know 3 of the sides. The Law of Cosine just doesn't seem, at least in an obvious way, that it's going to help me. I could also try to find this angle. Once again, we don't know all 3 sides to be able to solve for the angle. So maybe Law of Sines could be useful. So the Law of Sines, the Law of Sines. Let's say that this is, the measure of this angle is a, the measure of this angle is lower case b, the measure of this angle is lower case c, length of this side is capital C, length of this side is capital A, length of this side is capital B. The Law of Sine tells us the ratio between the sine of each of these angles and the length of the opposite side is constant. So sine of lower case a over capital A is the same as lower case b over capital B, which is going to be the same as lower case c over capital C. Let's see if we can leverage that somehow right over here. We know this angle and the opposite side so we can write that ratio. Sine of 40 degrees over 30. Let's see. Can we say that that's going to be equal to the sine of this angle over that? Well it would be, but we don't know either of these so that doesn't seem like it's going to help us. But, we do know this side. Maybe we could use the Law of Cosines to figure out this angle, because if we know 2 angles of a triangle, then we can figure out the third angle. So let's do that. Let's say that this angle right over here is theta. We know this distance right over here is 40 meters, so we can say that the sine of theta over 40, this ratio is going to be the same as the sine of 40 over 30. Now we can just solve for theta. Multiplying both sides times 40, you're going to get, let's see. 40 divided by 30 is 4/3. 4/3 sine of 40 degrees is equal to sine of theta, is equal to sine of theta. Now to solve for theta, we just need to take the inverse sine of both sides. So inverse sine of 4 over 3 sine of 40 degrees. Put some parentheses here, is equal to theta. That will give us that angle here and we can use that information and this information to figure out the angle that we really care about. So, let's get a calculator out and see if we can calculate it. Let me just verify, I am in degree mode. Very important. All right, now I'm going to take the inverse sine of 4/3 times sine of 40 degrees, and that gets me, and I deserve a little bit of a drum roll, 58, well if we round to the nearest, let's just maintain our precision here. So 58.99 degrees roughly. This is approximately equal to 58.99 degrees. So, if that is 58.99 degrees, what is this one? It's going to 180 minus this angle's measure minus that angle's measure. Let's calculate that. It's going to 180 degrees minus this angle, so minus 40, minus the angle that we just figured out. Actually I can get all of our precision by just typing in second answer. So that just says our previous answer so I get all that precision there and so I get 81.01 degrees. So, if I want to round to the nearest, let's say I want to round to nearest hundredth of a degree, then I'd say 81.01 degrees. So this, this right over here, is approximately 81.01 degrees and we're done.