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Sine and cosine

Soo..... why do sine and cosine exist? They are essentially the same with a phase shift of pi/2. Could cosine not be replaced with sin(x-pi/2) ? Why do they both exist???
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Avatar for Xei
Xei
6
Shift of pi/2 actually. x -> pi/2-x to interchange.

Basically it's because they're both as common as each other and writing cosx is a lot easier than writing sin(pi/2 -x).
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Avatar for Small123
Small123
15
davidmarsh01
Soo..... why do sine and cosine exist? They are essentially the same with a phase shift of pi. Could cosine not be replaced with sin(x-pi/2) ? Why do they both exist???

To find the angles of triangles was the initial idea.
davidmarsh01
Why do they both exist???


Sin would be lonely without his little pal to keep him company.
Interesting question, and I would say "because it's very useful to have a new function instead of performing operations with sin and sin(x-pi/2)". I mean, d/dx cos x = -sin x, d/dx sin x = cos x; so calculus is "closed". And there are myriad identities involving cos and sin, which do perhaps lose their neatness if we replace cos x with sin(x-pi). Plus there's the elegant series definition, cos x = 1 - x^2/2! + x^4/4! - ...

Also, how do we define cos x in terms of sin x? Do we choose sin(pi/2-x), sin(x+pi/2), 1−sin⁡2x\sqrt{1 - \sin^2 x}, or an infinite number of alternatives?

Why can't we just use sin(pi/2 - x)?

Be cos...

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