Thread: Exact values of sin20 and sin40 - impossible?

1. Exact values of sin20 and sin40 - impossible?

I'm trying to find exact trigonometric values. Say I have sin11, I would use the special ratios:

$sin11=sin(45-34)= etc. etc.$ and I would arrive with a surd answer, which is fine.

But I'm having trouble with angles like 20 and 40. Is there a reason for this?

2. Originally Posted by RAz
I'm trying to find exact trigonometric values. Say I have sin11, I would use the special ratios:

$sin11=sin(45-34)= etc. etc.$ and I would arrive with a surd answer, which is fine.

But I'm having trouble with angles like 20 and 40. Is there a reason for this?
Most of the are difficult to find.

For example $\sin(20^{\circ})=\sin\left(\frac{\pi}{9}\right)$. One can use this to get $\frac{\sqrt{3}}{2}\sin\left(\frac{\pi}{3}\right)=\ sin\left(\frac{\pi}{9}\cdot 3\right)=3\sin\left(\frac{\pi}{9}\right)-4\sin^3\left(\frac{\pi}{9}\right)$.

Thus, $\sin\left(\frac{\pi}{9}\right)$ is the solution to the depressed cubic $x^3+px=q$....I did something like this somewhere, hold on....ah! look here. That'll give you a taste of what solving them is like.

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how can we found value of sin20

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