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Don't get stuck trying to find missing sides or angles in a triangle. Use the Law of Sines to rescue you from any perilous triangle in which you have just a few pieces of information.

Conquering Triangles

Triangles aren't so scary, right? I guess it depends on context. The triangles of cheese that a sandwich shop uses are a welcome sight. The triangle on a pool table can be good or bad, depending on your pool shark-um-ness. And then there's the mysterious Bermuda Triangle, which I guess is kind of scary.

But right triangles are your friends.

If you see this one, where you know two sides and want the third, what do you do? Pythagorean Theorem! This tells us that the sum of the squares of the two legs equals the square of the hypotenuse, or a2 + b2 = c2.

And what about this one? You know this angle and the hypotenuse and need this side. Just use 'SOH CAH TOA', our awesome acronym to remember sine, cosine, and tangent. The SOH means that the sine of an angle equals the side opposite the angle over the triangle's hypotenuse.

This is the opposite side from the angle, so we say sine 30 equals the opposite over the hypotenuse, or x over 6. No problem!

Law of Sines

Unfortunately, we don't always have right triangles. Sometimes we have wrong triangles.

Ok, a triangle like this isn't wrong. And, in fact, we can still use sine to help us. And not just 'SOH CAH TOA', but something called the Law of Sines. This law has nothing to do with where you can and can't post signs. No, it's a super-helpful tool for tackling any triangle.

It goes like this: a/sin A = b/sin B = c/sin C. With our triangle, the capital A, B and C are the angles. And the lowercase a, b and c are the lengths of these sides opposite their respective angles.

With the Law of Sines, we're comparing ratios. The ratios of the sides to the sines are equal to each other.

Again, this works with any triangle. And you'll never need to use all three sides and angles. You only need two. So if you know two angles and one side, you can use the Law of Sines to find the missing side.

Likewise, if you know two sides and an angle, you can use the Law of Sines to find the missing angle, but we're just going to focus on finding a missing side here.

Practice Problem #1

Let's practice that first type of problem, when we know two sides and an angle. Here's a triangle.

We're told that angle A is 82 degrees and angle B is 20 degrees. We also know that side b is 6 units long. We can use the Law of Sines to find the length of side a.

Remember the law? It's a/sin A = b/sin B = c/sin C. We don't need that last part. Let's just plug in what we know. So it's a/sin 82 = 6/sin 20. The sine of 82 is .99, or almost 1. The sine of 20 is about .34. Let's cross multiply. 1 times 6 is 6. a times .34 is .34a. 6 divided by .34 is about 18. So side a is about 18 units long.

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Notice that the Law of Sines is really about cross multiplying. You're just taking the given information and plugging it into the law.

Practice Problem #2

Let's do another like this. In this triangle, we know that angle A is 100 degrees and angle B is 46 degrees. Also, we know that side a is 21 units long. What is the length of side b?

Let's use the Law of Sines! We have 21/sin 100 = b/sin 46. The sine of 100 is about .98. The sine of 46 is about .72. Now we cross multiply. 21 times .72 is about 15. 15 divided by .98 is, well, about 15. So side b is about 15 units long.

Ok, double check time. If side a is 21, does it look like side b could be 15? Yeah, I think so.

Practice Problem #3

Let's do one more. In this triangle, angle A is 65 degrees and angle B is 61 degrees. If side a is 42 units long, what is side b?

Back to the Law of Sines! We have 42/sin 65 = b/sin 61. The sine of 65 is about .91. The sine of 61 is about .87. We multiply 42 by .87 to get about 37. And 37 divided by .91 is about 41. So side b is about 41 units long.

Let's double check this. If angle B is just 4 degrees smaller than angle A, does it seem logical that side b is just barely shorter than side a? Yep, totally.

Lesson Summary

In summary, the Law of Sines is a wonderfully helpful tool that uses ratios to find missing sides and angles in any triangle.

The Law of Sines is a/sin A = b/sin B = c/sin C. The letters on the bottom represent the angles of the triangle. The letters on the top represent the lengths of the sides opposite those angles.

When using the law of sines, just plug in the info you know and cross multiply. And fear triangles no more.

Learning Outcome

Subsequent to watching and studying this lesson, you might be able to apply the Law of Sines to help find the missing sides or angles of a triangle. You could also understand the meaning of and utilize SOH CAH TOA.

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