# Exact value of trig function

#### jacobi

##### Active member
Find the exact value of $$\displaystyle \sin \frac{\pi}{180}$$.

#### topsquark

##### Well-known member
MHB Math Helper
Find the exact value of $$\displaystyle \sin \frac{\pi}{180}$$.
Ummmmm....is the argument in radians or degrees? Kinda hard to tell given the fraction.

-Dan

#### Plato

##### Well-known member
MHB Math Helper
Find the exact value of $$\displaystyle \sin \frac{\pi}{180}$$.

#### Opalg

##### MHB Oldtimer
Staff member
Find the exact value of $$\displaystyle \sin \frac{\pi}{180}$$ (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.

Last edited:

#### Fernando Revilla

##### Well-known member
MHB Math Helper
Related with the OP (but more simple): sin 72º.

#### jacobi

##### Active member
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.

#### jacobi

##### Active member
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.
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 $$\displaystyle \sqrt[3]{ \frac{a+b}{4}}$$, where $$\displaystyle a=\sqrt{8+\sqrt{15}+\sqrt{3}+\sqrt{10-2 \sqrt{5}}}$$ and $$\displaystyle b=\sqrt{-8+\sqrt{15}+\sqrt{3}+\sqrt{10-2 \sqrt{5}}}$$. Conversely, the cosine of $$\displaystyle 1^\circ$$ can be written as the real part of the above.

#### Prove It

##### Well-known member
MHB Math Helper
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*}