# Trigonometry : Law of Cosines

## Example Questions

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### Example Question #1 : Law Of Cosines

Given the measures of the sides as:  and , determine the measure of

Explanation:

Here we are given Side - Side - Side, therefore we must utilize the Law of Cosines. The angle we desire is , therefore, the equation must be setup such that the corresponding side is on the left hand side of the equals sign:

Substituting the given values:

Further reducing:

Dividing each side by 4:

Finally, we can utilize the inverse cosine function to determine the desired angle:

### Example Question #1 : Law Of Cosines

Which famous theorem does the Law of Cosines boil down to for right triangles?

Mean Value Theorem

Pythagorean Theorem

Cosine Theorem

Vertical Angle Theorem

Isosceles Triangle Theorem

Pythagorean Theorem

Explanation:

The Law of Cosines is as follows:

Notice these equations contains the Pythagorean Theorem, , within it.

The term at the end is the adjusting term for triangles which are not right triangles.

### Example Question #3 : Law Of Cosines

Find the value of  to the nearest tenth.

Explanation:

This is a prime example of a case that calls for using the Law of Cosines, which states

where , and  are the three sides of the triangle, and  is the angle opposite side .  Looking at our triangle, taking , then we have , and .  Plugging this into our formula, we get.

Using our calculator to approximate the cosine value gives

Simplifying further gives

Solving by taking the square root gives

### Example Question #4 : Law Of Cosines

Using the Law of Cosines, determine the perimeter of the above triangle.

Explanation:

To apply the Law of Cosines,  is the unknown,  and  are the respective given sides, and the given angle is .

Therefore, the equation becomes:

Which yields

Add  to the other two given sides to get the perimeter,

### Example Question #5 : Law Of Cosines

Solve for x:

Explanation:

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

In this case:

### Example Question #6 : Law Of Cosines

Find the missing angles and sides.

Explanation:

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°.

### Example Question #7 : Law Of Cosines

In the triangle below, , and . Find the measure of  to the nearest tenth.

There is not enough information.

Explanation:

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

### Example Question #2 : Law Of Cosines

In the triangle below,  meters, and  meters. What is the length of b, to the nearest tenth of a meter?

5.7 meters

There is not enough information.

8.5 meters

13.0 meters

9.0 meters

9.0 meters

Explanation:

The law of cosines states that .

So:

### Example Question #9 : Law Of Cosines

A radar tower detects two ships. Ship A is 730 meters away and  south of west. Ship B is 525 meters away and  north of west. What is the distance between the two ships to the nearest meter?

297 meters

899 meters

507 meters

696 meters

516 meters

696 meters

Explanation:

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:

### Example Question #3 : Law Of Cosines And Law Of Sines

In the triangle below, , and . What is the measure of  to the nearest tenth of a degree?

There is not enough information.

Explanation:

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:

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