why is sin(90-θ) = cosθ?

Question

why is sin(90-θ) = cosθ?

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Answer 1

Imagine a rectangle with a diagonal drawn bottom left to top right. The angle the diagonal makes with the horizontal base is theta and the length of the base is d.cos(theta) where d is the length of the diagonal. Adjacent over hypotenuse is cosine.

Now the angle the diagonal makes with the vertical side is (90 – theta) so the length of the top side is d.sin(90 – theta). Opposite over hypotenuse is sine.

But the length of the top is the same as the length of the base so:-

d.sin(90 – theta) = d.cos( theta) and the d can be cancelled out.