The Law of Sines can be used to find the angles and side lengths of triangles that aren't right triangles. It has little to do with crop circles, aliens, or Mel Gibson.
The triangle above represents any nonright triangle. The Law of Sines says that for such a triangle,
We can prove, too. All we have to do is cut that triangle in half.
Using the trig ratios we learned, we can find the sine of angles A and B for the right triangles we made.
Isolate for the altitude h and then set the two equations equal to each other.
b × sin A = h
a × sin B = h
b × sin A = a × sin B
Dividing both sides by ab will give us our Law of Sines.
If draw different altitudes and repeat all that business, we'll end up with and . Solid proof, we think. Now that we know it's true, we can start putting it to use.
Sample Problem
We'll begin by finding the value of x. The Law of Sines is still fresh in our minds.
We'll set A to 82° and B to 32°. That means a is 7 and b is x.
x ≈ 3.7
Simple enough.
The Law of Sines is so useful that with it, we're unstoppable. We can solve triangles, which means finding every side length and every angle measurement.
Sample Problem
Solving this triangle should be pretty easy. We'll start with the Law of Sines.
If A is 60°, then a is 4. The only other side we have is 3.2, so that'll be b.
We want B, not sin B. If we take the inverse sine of each side, we get:
B ≈ 43.9°
Wonderful. Now we have two out of three angles and two out of three sides. Knowing that all the angles in a triangle add up to 180° allows us to find the last angle.
A + B + C = 180°
Substitute in the angles we know.
60° + 43.9° + C = 180°
Solve for the remaining angle.
C = 180° – (60° + 43.9°)
C = 76.1°
One side to go. We can use either the ratio between sin(60°) and 4 or sin(43.9°) and 3.2.
Substitute in what we know. We're looking for the last side, c.
c ≈ 4.5
Doublecheck. Do we know all the angles and all the sides?
A = 60°; B = 43.9°; C = 76.1°
a = 4; b = 3.2; c = 4.5
Yep. We solved that triangle real good.
Practice:
Find x if ∆XYZ has a side y = 9.2 and angles of X = 38° and Y = 62°.  
This is a classic Law of Sines problem. The ratios are practically given to us outright. Insert our values. Solve for x. Plug it into a calculator. x ≈ 6.4 Right on.  
What is the perimeter, P, of this triangle?
 
Luckily, most of the work is done for us already. The perimeter is the sum of all the sides, and we already have two out of three given to us. All that's left is to find the last side using, you guessed it, the Law of Sines. If 70° is A, then a is 23 and b is 16. That means what we're solving for is B. Not what we're looking for, but we'll take it step by step. Solve for B. Inverse trig it up. Calculate. B ≈ 40.8° Now we have another angle. Since we have two of them, we might as well go ahead and find the last one. That should help us. A + B + C = 180° 70°+ 40.8° + C = 180° C = 69.2° We can use the Law of Sines one more time with the angle we just calculated. If all goes well, we should find our last remaining side. It doesn't really matter which angleside pair we use as long as we're consistent.
Rearrange to find c. Since 70° and 69.2° are really close, c shouldn't be too different from 23. c ≈ 22.9 We did it. Well, almost. To calculate the perimeter, we just have to add up all the sides. P = a + b + c P = 23 + 16 + 22.9 P = 61.9 Now we did it.  
Solve this triangle.


If we're given two angle measurements right off the bat, we can find
the third one no problem. We know that all the angles add up to 180°. A + B + C = 180° 20° + 80° + C = 180° Solve for the last guy. C = 80° Since both B and C
are 80°, this triangle is isosceles. Good to know, since that means the
remaining two sides are equal in length. We can use the Law of Sines. Substitute in the angles and the one side we do know. Solve for b. Use a calculator. b ≈ 5.8 Aww yeah. We've solved the triangle. A = 20°; B = C = 80° a = 2; b = c = 5.8  
Solve this triangle.
 
Two out of three angles? We know what to do. A + B + C = 180° 79° + 31° + C = 180° C = 60° All that remains is to find the sides of this unruly triangle. Thank goodness we have the law on our side. The Law of Sines, that is. Since C is 60°, we'll make A equal 79°. That leaves a unknown and 27 as c. Solving for the unknown. Plug that whole thing into a calculator and see what comes out. a ≈ 30.6 One more side and then we're done.
b ≈ 16.1
Triangle solved.  
Find the value t of ∆TUV where ∠T = 54°, ∠U = 67°, and u = 14.
Hint
The letter of the angle relates to the letter of the side. Using that and the Law of Sines, we should be good to go.
Find the value of x.
Hint
The angles of a triangle always add up to 180°.
Find the value of x.
Hint
Law of Sines, all the way.
Solve the triangle.
Hint
First, we can use the Law of Sines and our mad algebra skills to find x. Then, finding the sides should be a cinch.
Answer
A = 87°; B = 55°; C = 38°; a ≈ 27.9; b ≈ 22.9; c = x ≈ 17.2
Will, Bill, and Steve are playing baseball. Will, the pitcher, stands 60.5 ft away from Bill at home plate and 50 ft away from first base, where Steve is tying his shoe. If home makes a 44° angle with first, how far away are Bill and Steve, approximately?
Hint
Drawing this out might help. The angle and side ratio that goes together is the 44° angle at home and the 50 ft between Will and Steve. Finding all the angles is the best way to get to the answer.
Answer
Bill and Steve are about 85.7 ft apart.
A spaceship has traveled some distance away from its home planet, Zargon, along a straight path. Alluron, a nearby star, is behind the spaceship at an angle of 32°. If Zargon and Alluron make a 73° angle and are known to be 25 million lightyears away, how far has the spaceship traveled, in lightyears?
Hint
The spaceship, Zargon, and Alluron are three points of a triangle. If we abbreviate them S, Z, and A, we have ZA = 25, ∠AZS = 73°, and ∠ZSA = 32°. We have to find SZ.
Answer
The spaceship is about 43.7 million lightyears away from Zargon.
The roof of a house makes an isosceles triangle with a vertex of 50°. If each side of the roof is 18 feet long, how wide is the house?
Hint
The triangle is isosceles, which means that the two remaining angles must be equal to each other.
Answer
The house is about 15.2 feet wide.