Question

In right triangle ABC, m∠B ≠ m∠C. Let sin B = r and cos B = s. What is sin C – cos C?

Answer 1

Answer:   s - r

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Explanation:

We'll need these trig ratios

• sine = opposite/hypotenuse
• cosine = adjacent/hypotenuse

Refer to the diagram below.

sin(B) = r = r/1 which tells us that the side opposite angle B is r units long. The hypotenuse is 1.

cos(B) = s = s/1 means the adjacent side to angle B is of length s

sides r and s are the two legs of the right triangle.

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From that diagram, we can then say:

sin(angle) = opposite/hypotenuse

sin(C) = s/1

sin(C) = s

This is identical to cos(B). We can say cos(B) = sin(C)

Also, we can say:

cos(angle) = adjacent/hypotenuse

cos(C) = r/1

cos(C) = r

So cos(C) = sin(B)

This works because B+C = 90

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So,

sin(C) - cos(C) = cos(B) - sin(B) = s - r