username@email.com
username@email.com
In this lesson, you will learn how to apply the Law of Sines and the Law of Cosines to determine the length of sides and measure of angles for triangles other than right triangles.
.
.
.The Law of Sines and the Law of Cosines are used to determine the length of sides or measure of angles of triangles that are not right triangles.
Let A, B, and C represent the three interior angles of any triangle. Let a denote the length of the side that is opposite the angle A, b denote the length of the side that is opposite the angle B, and c denote the length of the side that is opposite the angle C.
The Law of Sines states that the ratio of two sides is equal to the ratio of the sines of their opposite angles. So the formula for the Law of Sines is
The Law of Sines can be used to solve for all of the angles of a triangle if all of the sides of the triangle are known. The Law of Sines can also be used to help solve for lengths of sides and measurements of angles. If you know two sides and one angle, you can find the length of the third side.
Make sure you pay attention to the definition and formula for the Law of Sines because you will be asked prove the Law of Sines at the end of this module.
What is the length of side a in the triangle on the right?
The correct choice is B. If we use the Law of Sines, we know that
.
By using cross-multiplication, we can also express the Law of Sines as
.
Therefore, 
.
So, the correct answer is 
.
The Law of Cosines is used when we know the lengths of sides a and b, and we also know the measurement of the angle between sides a and b, which as you can see in the triangle below, is angle C. Look at the triangle with sides a, b, and c. The side h is perpendicular to side a, and is included to split the triangle into two right triangles. Remember that splitting a triangle that is not a right triangle into more than one right triangle (through the use of a line that is perpendicular to a side of the existing triangle) is a tool that you have that will enable you to apply trigonometric formulas in cases where, at first glance, you may have thought you could not apply them.
The Law of Cosines can be shown as a formula by 
.
Using the Pythagorean Theorem, we know 
, which is expressed in expanded form as 
.
Also using the Pythagorean Theorem, we know that 
.
By substituting 
for 
, we get 
.
At this point, we have solved for sides a and b, and for angle C. We cannot stop here because the variable d must be solved for in order to finish the equation. Since we can choose to look at only one of the right triangles, we can apply one of our trigonometric ratios that we learned earlier. Do you remember soh cah toa? Since we are trying to find an equation for d, we know that we need to look at a trigonometric ratio that includes an adjacent side. Therefore, we are looking at either the “C-A-H” to get 
, or the “T-O-A” to get 
. Now we have sides, b and h in the triangle to look at. Since we do not know side h, we should look at side b. Side b is the hypotenuse in the right triangle with sides b, d, and h. So we know to use the trigonometric ratio .
Therefore, since we are attempting to solve for d, we need to multiply both sides by the hypotenuse b to get 
.
Now we can plug in 
for d in the equation 
. This gives us 
.
This proves the Law of Cosines, 
.
Using the Law of Cosines and the triangle on the right, what is the value of c?
The correct choice is B. Using the Law of Cosines, we know that 
.
Since a = 10, b = 10, and C = 60 ° , we can use substitution.
You can simply surmise by looking at the triangle that c = 10 m, because the triangle above is an equilateral triangle. Equilateral triangles, as we have learned, have all sides of equal length and all equal angle measurements (angles in an equilateral triangle are always 60° acute angles). The Law of cosines may not have been the easiest way to solve this particular problem, but it is nonetheless important to completely understand the concept.
Since you have tried an example that required use of the law of sines and an example that illustrated the law of cosines, we will try an example that uses both the law of sines and the law of cosines.
The correct choice is C. We know from the Law of Sines that 
.
First, we must solve for each missing variable.
Remember the rule that says, in order to solve for A, you can say that
.
Therefore, 
, and since the interior angles of a triangle adds up to 180°,
.
Using the Law of Cosines, we can say that
It is easiest to rotate the triangle as shown above.
The area of a triangle is 
.
Since we know the base, we only need to solve for the height in order to find the area.
Using the Law of Sines, we can determine that
Therefore, the area of the triangle is 
.
