Answer:
the answer is below
Explanation:
Let’s do it be an example, then all you have to do is substitute your numbers in place of the example numbers.
Let’s assume your triangle ABC has angle A = 20 degrees, angle B = 32 degrees, and angle C = 128 degrees. and side AB = 12.
We want to find the length of side AC and BC.
Side AB is opposite angle C
Side AC is opposite angle B
Side BC is opposite angle A
We are going to use the Sine Rule that says AB/sin C = AC/sin B = BC/sin A (or you can flip them all over and it will be just as good.
To use the rule, you need to know three of the parts of two of the fractions as a time.
Since 23 know side AB = 12, and the angle opposite it, angle C = 128 degrees, we will use that fraction and the fraction for side AC, next. For that fraction we need the measure of angle B = 32 degrees, then
AC/sin B = AB/sin C, then AC/sin(32) = 12/sin(128)
We will use a scientific calculator to find the decimal values for the sin(B), sin(C) and later sin(A).
sin(B) = sin(32) ≈ 0.5299, sin(C) ≈ sin(128) ≈ 0.7880
Then AC/0.5299 ≈ 12/0.7880, then AC ≈ (12)(0.5299)/(0.7880) ≈ (6.3588)/(0.7880), then
AC ≈ 8.0695
Now to find BC, we use BC/sin(A) = AB/sin(C) (Note we will be using the same fraction as before.)
Back to the calculator sin(A) = sin(20) = 0.3420
Then BC/sin(A) = AB/sin(C), then BC/0.3420 ≈ 12/0.7880, then BC ≈ (12)(0.3420)/(0.7880), then
BC ≈ 5.2081
Remember each fraction is a side divided by the sin of the angle “opposite” that side.
The Sine Rule can be used when you know two angles and the side opposite one of them, because with the two angles and the side opposite one of them to find the side opposite the other angle. Then with the two angles you can find the third angle and then the rule to find the last side oppoite that angle.
Or if you know two sides and one angle opposite one of the given sides. You use them to find the second angle opposite the other side. Then you can compute the third angle and use it to find the side opposite it.