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trigonometric functions of an angle

$\sin(\theta +2\pi )=\sin(\theta ),\qquad \cos(\theta +2\pi )=\cos(\theta ).$

${\textstyle \sin(\theta )=\cos \left(\theta -{\frac {\pi }{2}}\right)}$

$\sin(\arcsin(x))=x\qquad \cos(\arccos(x))=x$

$(\cos \theta ,\sin \theta )$

In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, … Wikipedia

Factsheet

General definition $\text{[math]}$

Fields of application Trigonometry, Fourier series, Mathematical analysis.

Factsheet

General definition $\text{[math]}$

Fields of application Trigonometry, Fourier series, Mathematical analysis.

Wikipedia

en.wikipedia.org › wiki › Sine_and_cosine

Sine and cosine - Wikipedia

6 days ago - In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the ...

Elementary descriptions Analytic descriptions Complex numbers relationship Background Software implementations

reddit.com › r/mathematics › what is sine?

r/mathematics on Reddit: What is sine?

April 28, 2021 -

So I get that Sin, Cos and Tan are used to find angles in a triangle using the length of sides, but what’s the equation behind the function? i.e. how does sin(90) become 1? What’s the series of calculations that have to be done?

In the way that to go from 10 to 200 you multiply 10 by 20, how do you get from sin(90) to 1?

Top answer

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Sine is best defined visually in my opinion using the unit circle. However, there is an equation but it works using angles in radians rather than degrees, and technically goes forever. sin(θ) = θ - ( θ³ / 3! ) + (θ⁵ / 5!) - (θ⁷ / 7!) + (θ⁹ / 9!) ... [Also 3! is three factorial and 3! = 1x2x3 = 6, 5! = 1x2x3x4x5 = 120, etc] To get from sin(90°) = 1, we have to first turn 90 degrees into radians. A full circle is 360 degrees, or 2π radians. So 90 degrees becomes 2π/4 = π/2 Then put it into the infinite sum: sin(90°) = π/2 - ( (π/2)³ / 3! ) + ( (π/2)⁵ / 5!) - ( (π/2)⁷ / 7!) + ( (π/2)⁹ / 9!) sin(90°) = π/2 - ( (π³/8) / 6 ) + ( (π⁵/32) / 120) - ( (π⁷/128) / 5040) + ( (π⁹/512) / 362880) ... sin(90°) = π/2 - ( π³ / 48 ) + ( π⁵ / 3480 ) - ( π⁷ / 645120) + ( π⁹ / 185794560) ... sin(90°) = π/2 - ( 31.006 / 48 ) + ( 306.020 / 3480 ) - ( 3020.293 / 645120) + ( 29809.099 / 185794560) ... sin(90°) = 1.570796 - 0.645964 + 0.079692 - 0.004682 + 0.000160 sin(90°) = 1.000002, with errors because i didn't do all infinite terms.

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Initially when studying mathematics, functions are all pretty simple. x2 + 1 means "take x things x times and add 1". There's an easy procedure to produce the value. But once you study more math you get to functions that have much more involved definitions, some of them requiring many nontrivial steps to compute. Sine is one of those. And it doesn't even stop there. Once you get to abstract algebra, you work with functions that you don't care how, and often don't even know how to compute the value of. You might have just defined them as "some function that solves so-and-so differential equation" and as long as you can prove one exists, for your purposes, you might not ever need to figure out how to actually compute the values it takes. What I'm getting at is this: Functions are precisely fixed mappings from some domain to some range. That's it. No more, no less. Sine is one such. There are some ways to compute its exact value at certain discrete points. But for the vast majority (infinitely many) of numbers x, sin(x) has to be approximated. We do have schemes that can approximate arbitrarily well if you run them for long enough, and computers are really good at just that. But largely we don't know the exact values of sin(x) for arbitrary points x. We approximate them as necessary and that's good enough. It is easy to incorrectly ascribe other properties to functions when you learn mathematics because most functions you encounter early on share some additional properties. But for something to be a function you actually don't need to know anything about it's range, domain and the values it takes in those, as long as you know there is some range, some domain and that the same input always gives the same output. I hope in the light of this, the first paragraph of the wikipedia article on functions makes sense to you.

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