Education.com
Try
Brainzy
Try
Plus

# The Law of Sines Study Guide (page 2)

(not rated)
By
Updated on Oct 2, 2011

## Finding Angles with the Law of Sines

We can use the Law of Sines to find any side of a triangle if we know two angles and one side. If we try to find the measure of an angle given two sides and one angle, however, a problem arises. For example, suppose we have the triangle in Figure 17.21. What is the measure of angle x?

Because we have two pairs of opposite angles and sides, we can set up the Law of Sines:

Our instinct is to evaluate x with the inverse sine function:

If we know that Figure 17.21 is drawn accurately, then this is the measure of angle x. However, if Figure 17.21 is not drawn accurately, then it is possible that the triangle is really the one in Figure 17.22.

In this case, x is an obtuse angle (greater than 90°) and thus could not be 70°.

In Figure 17.22, the measure of angle x is 110°, the supplement of 70°. This is because sin(110°) = sin(70°) as discussed after Figure 17.6:

sin(180° – θ) = sin(θ)

The inverse sine function will output angles only between –90° and 90°. That is to say, it only outputs acute angles. If we are seeking the measure of an acute angle θ with sin(θ) = r for a specific positive ratio r, then θ= sin–1(r). If we want the measure of an obtuse angle θ with sin(θ) = r, then θ = 180° – sin –1(r).

#### Example 1

Find the measure of the obtuse angle x in Figure 17.23.

We have x opposite the side of length 10 inches and 32° opposite the side of length 5.6 inches. Thus, we can use the Law of Sines:

so,

Using the inverse sine function, we get

sin–1(0.9463) ≈ 71.1°

This cannot be the measure of angle x because x is obtuse. Thus, x must be the supplement of this angle.

x ≈ 180° – 71.1° =108.9°

#### Example 2

Suppose the triangle in Figure 17.24 is drawn accurately. Find the measure of angle x.

Here, we use the Law of Sines:

Now:

sin–1(0.8631) ≈ 59.7°

Because x is an acute angle, this measure is correct: x ≈ 59.7°. If x had been obtuse, then its measure would be 180° – 59.7° = 120.3°.

Practice problems for this study guide can be found at:

The Law of Sines Practice Questions

View Full Article
Add your own comment

### Ask a Question

Have questions about this article or topic? Ask
150 Characters allowed

### Related Questions

#### Q:

See More Questions

### Today on Education.com

Top Worksheet Slideshows