[rock music]
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 allows 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 1/2 a b sine C,
notice C is the included angle for sides a 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 A B C 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
1/2 times the base times the height,
but if we take a look at this small right triangle
on the left, and in particular angle C,
the sine of angle C would be h over a,
and if we take this equation
and solve it for h,
but 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 it'd have the area as equal to 1/2 times b
times a sine angle C.
And rearranging this, we have one of the three
formulas that we just saw,
1/2 a b 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 six 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 would be equal to
1/2 times the product of the two sides
that include the angle,
so six 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.
So cross multiply,
divide by 35.2,
so we have the sine of angle A is approximately
equal to this quotient,
which is approximately .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 .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 1/2 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.
Okay, I hope you found this video helpful,
thank you and have a nice day.
[rock music]