[ROCK MUSIC] JAMES SOUSA: Welcome to a quick lesson on finding the area of a triangle using the sine function. And the goal is to determine the area of a triangle if given side, angle, side. 

So there are three formulas involving sine that allow us to find the area of a triangle. Remember that the relationship among the sides and the angles are as follows. Side A must be opposite angle A, side B must be opposite angle B, and side C must be opposite angle C. 

So when we take a look at this first formula, area equals one half AB sine C, notice C is the included angle for sides and B. And that's true for all three formulas. The angle we're using to find the value of sine must be the included angle of the given two sides. So that's the most important part about these formulas. The relationships are consistent. It's just based upon what information we're given in the problem. 

Let's take a look at where these formulas come from. If we have any triangle ABC, and construct an altitude from, let's say, vertex B so it's perpendicular to the opposite side, it forms two right triangles, a small one here and a larger one here. We know the area of any triangle is one half times the base times the height. But if we take a look at this small right triangle on the left, and a particular angle C, the sine of angle C would be h over a. And if we take this equation and solve it for h by multiplying both sides by a, we would have h is equal to a sine C. 

So we can take our original area formula and replace h with a sine c. So we'd have the area is equal to one half times b times a sine angle C. And rearranging this, we have one of the three formulas that we just saw. One half AB sine C will give us the area. 

So let's go ahead and take a look at a problem. Let's sketch a triangle, and let's label the given information. Angle A is 35 degrees, angle B is 82 degrees, side a is 6 centimeters, and side B is 15 centimeters. 

Remember, to use the area formula, we have to have the included angle. So we need to find the measure of angle C before we can apply that area formula. And we can do that because the sum of the interior angles must be 180 degrees. So angle C must be 63 degrees. This is the angle we need in order to apply that area formula because this is the included angle from the information we were given. 

So the area of this triangle will be equal to one half times the product of the two sides that include the angle. So 6 centimeters times 15 centimeters times the sine of the included angle, or the sine of 63 degrees. So now, we can find this product to find our area. So the area is approximately 40 square centimeters. 

Let's take a look at this problem now. Let's go ahead and sketch a triangle. Angle B is 72 degrees. Side A is 23.7 feet, and side B is 35.2 feet. Now remember, in order to use that area formula, we have to have the included angle. So we're going to have to find the length of side C to include angle B, or find the measure of angle C because it is included by side A and side B. 

And this becomes a little bit tricky because side C is opposite angle C, and we don't know either of those values. So let's find the measure of angle A, and then we can indirectly find the measure of angle C by using the law of sines. 

So the sine of angle A divided by 23.7 must equal the sine of angle B, which is 72 degrees divided by 35.2. Let's cross multiply, divide by 35.2. So we have the sine of angle A is approximately equal to this quotient, which is approximately 0.6403. So now, if we take the inverse sine of both sides, we'll have angle A is approximately equal to 40 degrees. 

Now remember, using the law of sines, this would fall into the ambiguous case. So there is another angle that has the sine function value of 0.6403 over in the second quadrant. That would be 140 degrees. But that would not be possible, since 140 plus 72 is more than 180 degrees. So we do only have one possible measurement for angle A. So we can label this 40 degrees. This is 40, and this is 72. That leaves 68 degrees for angle C. 

Now the 68-degree angle is included, and we know the length of side A and side B. So we can now use this information to find the area of that triangle. So the area is equal to one half times the product of the two sides times the sine of the included angle. So the area is approximately 387 square feet. 

So this is a nice way to find the area if you have the necessary information. And sometimes, you do have to find additional information in order to use it. OK, I hope you found this video helpful. Thank you, and have a nice day. 

[ROCK MUSIC]