How to solve $\sin(x) + 2 \sin (2x) = 0.74$?

Question

I cannot solve this simple looking trigonometric equation : $\sin(x) + 2\sin(2x) = 0.74$ I had searched the browser but cannot get a similar question. Personally I can solve the same question when its presented as: $\sin(x)+2\sin(2x) = 0$. May I get help on the above?

Answer 1

As Michael Rozenberg commented, this problem leads to a quartic equation $$\sin(x)+2\sin(2x)=a\implies \sin(x)+4\sin(x)\cos(x)=a\implies\cos(x)=\frac{a-\sin(x)}{4\sin(x)}$$ Square both sides $$\cos^2(x)=\left(\frac{a-\sin(x)}{4\sin(x)} \right)^2=1-\sin(x)^2$$ Let $t=\sin(x)$ to get $$1-t^2=\left(\frac{a-t}{4t} \right)^2\implies 16 t^4-15 t^2-2 a t+a^2=0$$ which is solvable using radicals but the solution would be very ugly and it should simpler to use a numerical method.

If you plot the function $$f(x)=\sin(x)+2\sin(2x)-0.74$$ you will notice that, in the range $0 \leq x \leq 2\pi$ , its zeros are close to $0$, $1.5$, $3.5$ and $4$. So, use Newton method using these as starting guesses. As iterates, you should get

For the first root $$\left( \begin{array}{cc} 0 & 0.000000 \\ 1 & 0.148000 \\ 2 & 0.149900 \\ 3 & 0.149901 \\ 4 & 0.149901 \end{array} \right)$$ For the second root $$\left( \begin{array}{cc} 0 & 1.50000 \\ 1 & 1.63878 \\ 2 & 1.63545 \end{array} \right)$$ For the third root $$\left( \begin{array}{cc} 0 & 3.50000 \\ 1 & 3.39265 \\ 2 & 3.40284 \\ 3 & 3.40292 \end{array} \right)$$ For the fourth root $$\left( \begin{array}{cc} 0 & 4.00000 \\ 1 & 4.39001 \\ 2 & 4.25156 \\ 3 & 4.23672 \\ 4 & 4.23650 \end{array} \right)$$ which are the solutions for six significant figures.