SOLUTION: If 0° < A, B < 90°, and sinA = cosB, compute cot(A+B)

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Question 1152726: If 0° < A, B < 90°, and sinA = cosB, compute cot(A+B)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

sin%28A%29+=+cos%28B%29 if and only if A%2BB+=+90. We say that angles A and B are complementary

For example, sin%2840%29+=+cos%2850%29 because 40%2B50+=+90
Using a calculator, sin%2840%29+=+0.6427876 and cos%2850%29+=+0.6427876 (both approximate).

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Because we are working with cotangent, let's start with the identity for tangent and rewrite it in terms of cotangent.

tan%28x%29+=+sin%28x%29%2Fcos%28x%29

tan%28x%29%2F1+=+sin%28x%29%2Fcos%28x%29

1%2Ftan%28x%29+=+cos%28x%29%2Fsin%28x%29 Apply reciprocal to both sides.

cot%28x%29+=+cos%28x%29%2Fsin%28x%29 We will use this later. Call this equation (1).

It might be tempting to use the reciprocal identity cot%28x%29+=+1%2F%28tan%28x%29%29 but if x+=+90 degrees, then tan%28x%29+=+0, making 1%2F%28tan%28x%29%29 to be undefined. Instead it is better to use cot%28x%29+=+cos%28x%29%2Fsin%28x%29

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With that in mind, we can compute the cotangent value.

cot%28A%2BB%29+=+cot%2890%29 Replace A+B with 90.

cot%28A%2BB%29+=+cos%2890%29%2Fsin%2890%29 Use equation (1).

cot%28A%2BB%29+=+0%2F1 Use the unit circle to compute cos(90) and sin(90).

cot%28A%2BB%29+=+0 Final answer.