# Trigonometry : Law of Sines

## Example Questions

### Example Question #1 : Law Of Sines

Given sides  and angle  determine the corresponding value for

Undefined

Explanation:

The Law of Sines is used here since we have Side - Angle - Side. We setup our equation as follows:

Next, we substitute the known values:

Now we cross multiply:

Divide by 10 on both sides:

Finally taking the inverse sine to obtain the desired angle:

### Example Question #1 : Law Of Sines

Let   and , determine the length of side .

Explanation:

We have two angles and one side, however we do not have . We can determine the angle using the property of angles in a triangle summing to :

Now we can simply utilize the Law of Sines:

Cross multiply and divide:

Reducing to obtain the final solution:

### Example Question #1 : Law Of Sines

In the above triangle,  and . If , what is  to the nearest tenth? (note: triangle not to scale)

Explanation:

If we solve for , we can use the Law of Sines to find .

Since the sum of angles in a triangle equals ,

Now, using the Law of Sines:

### Example Question #37 : Law Of Cosines And Law Of Sines

By what factor is  larger than  in the triangle pictured above.

It isn't

Explanation:

The Law of Sines states

so for a and b, that sets up

### Example Question #38 : Law Of Cosines And Law Of Sines

Solve for :

Explanation:

To solve, use the law of sines, where a is the side across from the angle A, and b is the side across from the angle B.

cross-multiply

evaluate the right side

divide by 7

take the inverse sine

### Example Question #39 : Law Of Cosines And Law Of Sines

Evaluate using law of sines:

Explanation:

To solve, use law of sines, where side a is across from angle A, and side b is across from angle B.

In this case, we have a 90-degree angle across from x, but we don't currently know the angle across from the side length 7. We can figure out this angle by subtracting from :

Now we can set up and solve using law of sines:

cross-multiply

evaluate the sines

divide by 0.9063

### Example Question #40 : Law Of Cosines And Law Of Sines

What is the measure of  in  below? Round to the nearest tenth of a degree.

Explanation:

The law of sines tells us that , where ab, and c are the sides opposite of angles AB, and C. In , these ratios can be used to find :

### Example Question #1 : Law Of Sines

Find the length of the line segment  in the triangle below.

Round to the nearest hundredth of a centimeter.

Explanation:

The law of sines states that

.

In this triangle, we are looking for the side length c, and we are given angle A, angle B, and side b. The sum of the interior angles of a triangle is ; using subtraction we find that angle C.

We can now form a proportion that includes only one unknown, c:

Solving for c, we find that

.

### Example Question #1 : Law Of Sines

In the triangle below, , and . What is the length of side  to the nearest tenth?

Explanation:

First, find . The sum of the interior angles of a triangle is , so , or .

Using this information, you can set up a proportion to find side b:

### Example Question #1 : Law Of Sines

In the triangle below, , and .

What is the length of side a to the nearest tenth?