MHB What is the exact value of the trig function for pi/180?

[jacobi1]

Find the exact value of $$\sin \frac{\pi}{180}$$.

 

[topsquark]

Find the exact value of $$\sin \frac{\pi}{180}$$.

Ummmmm...is the argument in radians or degrees? Kinda hard to tell given the fraction.

-Dan

 

[Plato2]

Find the exact value of $$\sin \frac{\pi}{180}$$.

Just ask the Wolf. HERE

 

[Opalg]

Find the exact value of $$\sin \frac{\pi}{180}$$ (in other words, $\color{red}{\sin 1^\circ}$).

Start with the known formula for $\sin 3^\circ$ (you can find it here): $\sin 3^\circ = \frac1{16}\Bigl(2(1-\sqrt3)\sqrt{5+\sqrt5} + \sqrt2(\sqrt5-1)(\sqrt3+1) \Bigr).$ The formula $\sin(3\theta) = 3\sin\theta - 4\sin^3\theta$ then tells you that $\sin1^\circ$ is the smaller of the two positive roots of the cubic equation $$4x^3 - 3x + \tfrac1{16}\Bigl(2(1-\sqrt3)\sqrt{5+\sqrt5} + \sqrt2(\sqrt5-1)(\sqrt3+1) \Bigr) = 0,$$ which can be solved exactly, for example by Vieta's method. But don't expect a neat solution. (Emo)

 

[Fernando Revilla]

Related with the OP (but more simple): http://www.fernandorevilla.es/ii/vi/58-2.

 

[jacobi1]

Ummmmm...is the argument in radians or degrees? Kinda hard to tell given the fraction.

-Dan

By convention, if there is no degree symbol in the argument, it is in radians.

 

[jacobi1]

Start with the known formula for $\sin 3^\circ$ (you can find it here): $\sin 3^\circ = \frac1{16}\Bigl(2(1-\sqrt3)\sqrt{5+\sqrt5} + \sqrt2(\sqrt5-1)(\sqrt3+1) \Bigr).$ The formula $\sin(3\theta) = 3\sin\theta - 4\sin^3\theta$ then tells you that $\sin1^\circ$ is the smaller of the two positive roots of the cubic equation $$4x^3 - 3x + \tfrac1{16}\Bigl(2(1-\sqrt3)\sqrt{5+\sqrt5} + \sqrt2(\sqrt5-1)(\sqrt3+1) \Bigr) = 0,$$ which can be solved exactly, for example by Vieta's method. But don't expect a neat solution. (Emo)

My method was exactly the same as yours, but I went ahead and solved the cubic. I found that the smallest positive solution can be written as the imaginary part of $$\sqrt[3]{ \frac{a+b}{4}}$$, where $$a=\sqrt{8+\sqrt{15}+\sqrt{3}+\sqrt{10-2 \sqrt{5}}}$$ and $$b=\sqrt{-8+\sqrt{15}+\sqrt{3}+\sqrt{10-2 \sqrt{5}}}$$. Conversely, the cosine of $$1^\circ$$ can be written as the real part of the above.

 

[Prove It]

The exact value of \displaystyle \begin{align*} \sin{ \left( \frac{\pi}{180} \right) } \end{align*} IS \displaystyle \begin{align*} \sin{ \left( \frac{\pi}{180} \right) } \end{align*} :p