Law of Sines Examples

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 angle-side 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