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## Geometry (all content)

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

Lesson 9: The 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?

• does this law of sines work for all kinds of triangles? •   Yes, as long as none of the sides have length zero.

That's a great catch, by the way, to notice that Sal's proof here is assuming that you're dealing with an acute triangle. If, for instance, β were the measure of an obtuse angle, you'd have to draw a slightly different picture and you'd actually be showing that sin(180 - β) = x/A. But that doesn't damage the result because sin(180-θ) = sin(θ) for any angle, so the Law of Sines still holds.
• When you are using law of sine for a triangle that is SSA, you can get the "ambiguous case" where there are 2 possibilities for the degrees, etc of the triangle. If I have already found the first triangle, then how do I find the second triangle? •  First, it only shows up if the information you are given is 2 sides and 1 angle which is opposite one of the given sides.

When you write and solve the law of sines, you end up with sinC=0.32 or something. You type sin^-1(0.32) in your calculator and you are given an acute angle. Actually there are two solutions to the equation sinC=0.32. One is acute (your calculator gave it to you) and the other solution is obtuse. To find the other solution, do 180deg - acuteangleanswer.

The obtuse angle is the beginning of the second 'ambiguous' case, and you find the remaining unknown sides/angles using that obtuse angle.

So say you are given sides a and b and angle A. You will get two answers for angle B (one acute and one obtuse). Each of these answers leads to two different angle C's and side c's.

As a side note - to check what I said about acute/obtuse angles, type
sin(30) and sin(150) into your calculator - you will notice they are equal.
• 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?
• Anyone know if he did another video on him implementing the law of sines? • Does this work for an obtuse triangle as well? • 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
• What if you are trying to find the sine of the right angle in a triangle? That's opposite over hypotenuse, and the opposite side IS the hypotenuse. That should be one. Am I correct?

What if you are trying to find the cosine of the right angle in a triangle? In that case, the hypotenuse is clear, but which of the other two is the adjacent? I don't know how to tell because in the case of a right angle in the triangle, the opposite side is the hypotenuse.

Any help is much appreciated. :) • 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.
• 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!
• 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? 