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### Course: High school geometry (staging)>Unit 6

Lesson 1: Law of sines

# Proof of the law of sines

Sal gives a simple proof of the Law of sines. Created by Sal Khan.

## Want to join the conversation?

• How could you put the law of sine into a word problem?
• June wants to measure the distance of one side of a lake. The lake can be expressed as the triangle ABC. Angle a is opposite side BC, angle b is opposite side AC, and angle c is opposite side AB. She knows angle a= 54 degrees and angle b= 43 degrees. She also knows side AC= 106 feet. What is the measure of side BC?
• Does this work for an obtuse triangle as well?
• Yes, the law of sines works for any type of triangle!!
• Anyone know if he did another video on him implementing the law of sines?
• Unfortunately no, Sal does have one on Cosine, and Law of Cosines. For more help on the Law of Sines, you could check out IXL. Hope this helps
• does anyone know where i can find videos for the double angle, half angle and product- sum formulas on this website or any other place?
• If you consider a and h as both being x in the addition rules for sine and cosine, you can easily figure out the double angle formulas.
In other words:
sin(2x) = sin(x+x) = sinxcosx + cosxsinx = 2sinxcosx
and
cos(2x) = cos(x+x) = cosxcosx - sinxsinx = (cosx)^2 - (sinx)^2
• How do I know that I'm supposed to write the law of sines as sinA/a=sinB/b or as a/sinA=b/sinB?
• It works either way! But I like to arrange it so that the unknown value is in the numerator of the fraction to the left of the equal sign.

For example, if I don't know side b, I would write the equation like this:
b / sinB = a / sinA

Then it's really easy to rearrange the equation, plug in the values, and solve for b:
b = (a ⋅ sinB) / sinA
etc.

If I didn't know ∠A, I would write (and rearrange) the equation like this:
sinA / a = sinB / b
sinA = (a ⋅ sinB) / b
∠A = sin⁻¹ [(a ⋅ sinB) / b]
etc.

Hope this helps!
• if in trig, side b =26sin47 divided by sin32 how does b=35.9
• Cindy, 35.9 is a correct answer. Well, I tried solving this on my calculator, and I actually got 35.8831949. Its just that its rounded to the nearest tenths thats why instead of having it in 35.88, it becomes 35.9. You know there are some calculators that round off answers right away.
• why sin 77= 180-77?
like sin theta = 180-theta? just how?
• Hi,😄
*Let's make some recap first:-👍😁
You see, 180 degrees is the sum of all degrees together in triangles, and sine (sin) is a law of opposite/hypotenuse... and theta is a (greek) symbol used for unknown angle values,👍OK?

Now that we recapped, let's answer ur question :

*This video proofs that the law of sines is true, so basically, Sal is giving us this proof. Like any proof, we need to provide: example- different ways to solve that come with the same answer as the theory- and correct answer. Sal made a shortcut as proof. If you try with the calculator sine of any angle equals the same as subtracting the whole sum of angles from a given angle (variable or number).😁

Know that practice makes perfect, the law will make sense to you when you practice it with other problems or experiment it with a triangle.😄(Cool scientist glasses on 👩🏼‍🔬!!)

Hope I got to your point!!🤗

#YouKhanLearnAnything
• Is there a reason why the law of sines works? I mean why the triangle has opposite sides and angles in equal ratios? I do get how he derived it but was wondering why the triangle has angles and their opposite sides in equal ratios to other angles and their opposite sides?
• The derivation actually explains it.

If you have any triangle and are comparing two of its angles and their corresponing sides, so like alpha and A or Beta and B, you can ue the third angle to drop a perpendicular and form two right triangles. Then both right triangles have the one side x. Then it's just a matter of using algebra. so sin(alpha) = x/B and sin(beta) = x/A. So in less math, splitting a triangle into two right triangles makes it so that perpendicular equals both A * sin(beta) and B * sin(alpha). Then you can further rearange this to get the law of sines as we know it.

So to summarize, split a triangle in half with a line x and it makes two right triangles. As long as you know soh cah toa you can make a relationship relative to that new x line. And fromt here a relationship with each other.

Let me know if that did not help.
• why is the way you guys round so complicated, if i round the wrong way it makes it wrong and to be honest its really annoying