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Exact value of trig function

jacobi

Active member
May 22, 2013
58
Find the exact value of \(\displaystyle \sin \frac{\pi}{180}\).
 

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,123
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
Jan 27, 2012
196

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,703
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. (Emo)
 
Last edited:

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
Related with the OP (but more simple): sin 72º.
 

jacobi

Active member
May 22, 2013
58
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
May 22, 2013
58
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 \(\displaystyle \sqrt[3]{ \frac{a+b}{4}}\), 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
Jan 26, 2012
1,403
The exact value of [tex]\displaystyle \begin{align*} \sin{ \left( \frac{\pi}{180} \right) } \end{align*}[/tex] IS [tex]\displaystyle \begin{align*} \sin{ \left( \frac{\pi}{180} \right) } \end{align*}[/tex] :p