why is sin(90-θ) = cosθ?


why is sin(90-θ) = cosθ?

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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.

They’re both the same shape and they’re both repeating. The only difference is the start point, so if you move it, they both become the same thing.

Sin and cos are the y and x values, respectively, of a point on the unit circle. Because you can drop a vertical line from any point on the unit circle and form a right triangle, and because the formula of the unit circle is x^2 + y^2 = 1, you can also find another right triangle on the circle by flipping the x and y coordinates. But the two angles opposite the distances x and y are complementary (because the third angle of the triangle is always a right angle by the definition of how we are constructing the triangle), so the sin of one of them must be the cos of the other and vice versa.

Look at a right triangle. The side that is “opposite” for theta is “adjacent” for 90-theta, and vice versa.

There are many explanations. This is the dumbest one:

In a right triangle, the sine is defined as the opposite side divided by the hypothenuse. The cosine is the adjactent side divided by the hypothenuse.

If you take either angle and substract it from 90, you’ve basically calculated the remaining angle of the triangle (other than the right angle itself). That’s because the sum of angles in any triangle is 180°, so finding the unknown angle is equivalent to solving 180=x+90+θ for x. That yields x=90-θ.

However, from x’s perspective, the opposite side (to θ) is the adjacent side. And vice versa. So the cosine of x is just the sine of θ. And vice versa. That’s why sin(90-θ) is just the cos(θ).