We can solve for x using the law of cosines, where C is the angle between sides a and b.

In this case:

The Law of Cosines come in different forms depending on which angle or side you wish to find. One of the missing bits of information about our triangle is side length a. It is important to find this side because with side length a we can use the Law of Sines to easily find the angle measures. Side a "unlocks" the problem.

The pertinent LOC is .

Now that we know side a, we can use the reciprocal form of the Law of Sines to find the remaining angle measures.

Angle B:

To find the corresponding angle we take the inverse sine. 

But there are two angles between 0° and 180°; there is 44.7° and . How do we know which angle to choose? We find out by solving for the last angle C with both of our hypothetical angles for angle B. Since side c is the largest side, it follows it should have the largest angle of all three angles in the triangle. Compute the measure of angle C by subtracting the given angle (angle A) and the angle we calculated (angle B) from 180°. Do this once with 44.7° and once with 135.3°. The first case results in the largest angle C and fits with c being the largest side. Thus angle B=44.7° and angle C must equal 110.3°.

To find an angle in an oblique triangle where all sides are known, use the law of cosines:

The law of cosines states that .

So:

The sketch of the situation below shows that the angle between the ships from the radar station is 65 degrees.

To find the distance between the ships, use the law of cosines:

To find , you must first find side c using the law of cosines:

Knowing c, you can find  using the law of sines or the law of cosines.

Law of sines:

Law of cosines:

Recall the law of cosines to determine the length of one side  of a triangle  given the lengths of the other sides  and  and their included angle :

Here, the unknown side length is denoted , and the other sides and the included angle is given. Substitute these values into the law of cosines and estimate square roots to the nearest thousandth decimal place to determine the side length .

Hence, the length of the remaining side  of triangle  is approximately  units.

We will label the side with length 6 as , the side with length 7 as , the unknown side as , and the angle between them as .

 The law of cosines relates the three sides and the angle as follows:

.

Plugging in 6 for , 7 for , and 120 for  yields:

We then take the square root of both sides:

The law of cosines is a generalization of the Pythagorean Theorem:

where a, b, and c are lengths of sides and C is the angle opposite to c.

This problem gives us values for a, b, and C, so we can use the law of cosines.

After simplifying:

Taking the square root of both sides:

In order to find the value of the angle, you will need to use the law of cosines. Recall that for any triangle, like the one shown below,

Now, since we want to find the value of B, we will need to use .

Plug in the given values of the triangle.

 degrees

Make sure to round to  places after the decimal.