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Example Questions
Example Question #1 : Law Of Cosines
Which famous theorem does the Law of Cosines boil down to for right triangles?
Isosceles Triangle Theorem
Vertical Angle Theorem
Pythagorean Theorem
Cosine Theorem
Mean Value Theorem
Pythagorean Theorem
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 #1 : Law Of Cosines
Find the value of to the nearest tenth.
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 #1 : Law Of Cosines
Using the Law of Cosines, determine the perimeter of the above triangle.
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 #1 : Law Of Cosines
Solve for x:
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 #1 : Law Of Cosines And Law Of Sines
Find the missing angles and sides.
None of the other answers.
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 #6 : Law Of Cosines And Law Of Sines
In the triangle below, ,
, and
. Find the measure of
to the nearest tenth.
There is not enough information.
To find an angle in an oblique triangle where all sides are known, use the law of cosines:
Example Question #1 : 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.
9.0 meters
13.0 meters
8.5 meters
9.0 meters
The law of cosines states that .
So:
Example Question #1 : Law Of Cosines And Law Of Sines
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
507 meters
899 meters
696 meters
516 meters
696 meters
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 #9 : 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.
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:
Example Question #10 : Law Of Cosines And Law Of Sines
Given the triangle , where
,
, and
, calculate the side length
to the thousandth decimal point.
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.
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