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Can we find the exact value(not numerical/approximation) of sin 1? I tried to do so by solving a cubic equation using Cardano formula but I ended up with complex nested radicals): I was told to use other methods but that won't yield exact values

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Wouldn't its exact value be $\sin(1)$? –  AnonSubmitter85 Aug 25 '13 at 7:24
If the angle is in degree, have a look into –  lab bhattacharjee Aug 25 '13 at 7:25
Isnt sin(1) an irrational? –  al-Hwarizmi Aug 25 '13 at 7:26
I mean without computer software, how do you find it manually?Can you show me? –  Tom Lynd Aug 25 '13 at 7:27
@ al Hwarizmi yes it is but can't it be represented in a form like the multiples of 3 can be... –  Tom Lynd Aug 25 '13 at 7:29

3 Answers 3

sin(1) is a Transcendental number.

ie ,sin(1) is not a solution a non-zero polynomial equation with rational coefficients.In other words sin(1) can't be expressed in closed form only using fractions and radicals. Note: sin(a), where a is algebraic(not transcendental) ,is transcendental.

Does that help you?

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@TomLynd , I didn't really understand what you meant by their solutions being incorrect. The links are for info on transcendental number and for the reason why sin(1) is transcendental. What I meant by sin(1) being not the solution of...... is that sin(1) can't be a solution of a polynomial equation with non-zero rational coefficients. There is no closed expression which has only radicals,fractions and algebraic numbers. There might be some expressions which involve transcendental numbers like e and $\pi$. –  jaseem Aug 25 '13 at 12:35
$\sin (1) = \Im(e^i)$ –  Mats Granvik Aug 25 '13 at 12:51
@ jaseem you better have a look at – lab bhattacharjee –  Tom Lynd Aug 25 '13 at 13:55
@TomLynd that is for one degree , or for $sin(\frac{\pi}{180})$ and $\frac{\pi}{180}$ is not algebraic.But 1(1 radian) is algebraic and therefore sin(1) is transcendental. –  jaseem Aug 25 '13 at 15:04
Series[Sin[x], {x, 0, 14}]

$$\sin (x) = x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5040}+\frac{x^9}{362880}-\frac{x^{11}}{39916800}+\frac{x^{13}}{6227020800}+O\left(x^{15}\right)$$

DeleteCases[CoefficientList[Series[Sin[x], {x, 0, 14}], x], 0]

$$\sin (1) = 1-\frac{1}{6}+\frac{1}{120}-\frac{1}{5040}+\frac{1}{362880}-\frac{1}{39916800}+\frac{1}{6227020800}-...$$

$$1-\frac{1}{6}+\frac{1}{120}-\frac{1}{5040}+\frac{1}{362880}-\frac{1}{39916800}+\frac{1}{6227020800}-... = 0.84147098480866...$$

$$\sin (1) = 0.84147098480790...$$

Table[(2*n - 1)!, {n, 1, 7}]

{1, 6, 120, 5040, 362880, 39916800, 6227020800}

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He wants a closed form. My comment addresses this. –  jaseem Aug 25 '13 at 12:26

If $1$ is degrees here's 2 closed forms:

$$\frac{1}{2}ie^{-\frac{i\pi}{180}} - \frac{1}{2}ie^{\frac{i\pi}{180}}$$

$$-\frac{1}{2}(\sqrt[180]{-1} - 1)(\sqrt[180]{-1} + 1)(-1)^{\frac{89}{180}}$$

If $1$ is radians here's one closed form:

$$\frac{i e^{-i}}{2}-\frac{i e^i}{2}$$

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