How to find the exact angle by hand from given trigonometric function value?

Question

$$ \sin(\theta - \frac{3\pi} 5 ) = \frac {1}{35} $$

Is it possible to solve for $\theta$ by hand, without using the arc function on the calculator, to get an exact angle value?

Answer 1

For very small angles $x$, you have $\sin x \approx x$, so here since the RHS is small you get $$ \frac{1}{35} = \sin \left( \theta - \frac{3\pi}{5}\right) \approx \theta - \frac{3\pi}{5}, $$ which yields $$ \theta \approx \frac{1}{35} + \frac{3\pi}{5} = \frac{1+21\pi}{35}. $$

Wolfram Alpha evaluates $\arcsin(1/35) - 1/35 \approx 1.89 \times 10^{-6}$, so the approximation is very precise.

Answer 2

you can write $$-\frac{1}{4}\sqrt{5}\sin(\theta)+\frac{1}{4}\sin(\theta)-\sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}}\cos(\theta)=\frac{1}{35}$$ and then we can convert the functions $\sin$ and $\cos$ into $\tan$ but this makes the things complicated