What is the length of side c to the nearest whole number if side a 105 and angle A 65 degrees and angle B 37 degrees?

since A+B+C = 180, we can see that C = 78°

Now, use the law of sines:

c/sinC = a/sinA

To find the length of side c, we can use the law of sines, which states that the ratio of the length of a side of a triangle to the sine of its corresponding angle is the same for all three sides.

The law of sines can be written as: a/sin(A) = b/sin(B) = c/sin(C)

Given that side a = 105 and angles A = 65 degrees and B = 37 degrees, we can find the length of side c by using the formula:

c = (a * sin(C)) / sin(A)

First, we need to find angle C. Since the sum of angles in a triangle is 180 degrees, we can find angle C by subtracting angles A and B from 180:

C = 180 - A - B
C = 180 - 65 - 37
C = 78 degrees

Now we can substitute the known values into the formula:

c = (105 * sin(78)) / sin(65)

Using a calculator to find the sine values, we get:

c ≈ (105 * 0.97815) / 0.90631
c ≈ 102.606

Finally, rounding to the nearest whole number, side c is approximately equal to 103.