Is it impossible to compute $\frac{11\sin(52)}{\sin(116)} = ? $ without knowing the exact value of $\sin(52)$ and $\sin(116)$

Question

$$\frac{11\sin(52)}{\sin(116)} = ? $$

I wondered whether there exists such way that allows me to compute this quickly or easier. Could you share your dear thoughts with me?

EDIT:

Since $\sin (\pi - x ) = \sin (x)$ we have that

$$\frac{11\sin(52)}{\sin(116)} = \frac{11\sin(52)}{\sin(26)}$$

In this case, $52$ is $2$ times of $26$. We can use double angle identity.

Answer 1

Assuming you are using degrees, we have $\sin(116)=\sin (90+26)=\cos (26)$ Then $$\frac {11\sin (52)}{\sin (116)}=\frac {11\cdot2\sin (26)\cos (26)}{\cos (26)}=22\sin (26)$$ which is simpler but not a decimal answer. After this, you can use a Taylor series centered at $\sin(30)$ to get an approximate answer.

Answer 2

Note that

$$\sin(90+x) = \cos x \implies \sin (116) = \cos (26)$$

$$\implies \frac{11\sin (52)}{\cos (26)}$$

And recall $$\sin 2x = 2\sin x \cos x$$

which yields

$$\frac{22\sin(26)\cos(26)}{\cos (26)} = 22\sin{(26)}$$

You could estimate using the expansion for $\sin x $ perhaps, which doesn’t seem too practical here.

$$\sin x = x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+…$$

Using $x = \frac{13\pi}{90}$ (in radians) and evaluating the first two or three terms, you can get a decent approximation and multiply the value by $22$.