Inverse Function

Using a calculator, it will tell you that $\theta= 44.4^\circ$ . We notice that $\sin {\theta} = 0.7 \approx$ but $< \frac {1}{\sqrt{2}} = 0.7071$ , implying $\theta \approx$ but $< 45^\circ$ . Since it is an objective question, we find that the nearest answer is $44.4^\circ$ . But we are required to "find" or "calculate" it. I am giving a solution to "estimate" it.

Let $\theta = \frac {\pi}{4} - \phi$ and $\sin{\phi} = x$ ; then we have:

$\sin{\theta} = \sin{(\frac{\pi}{4} - \phi)} = \sin{\frac{\pi}{4}} \cos {\phi} - \cos {\frac{\pi}{4}} \sin {\phi} = \frac {1}{\sqrt{2}} \sqrt{1-x^2}-\frac {1}{\sqrt{2}}x = 0.7$

$\Rightarrow 10\sqrt{1-x^2}-10x = 7\sqrt{2} \quad \Rightarrow 10\sqrt{1-x^2} = 10x + 7\sqrt{2}$

$\Rightarrow 100 - 100 x^2 = 100 x^2 + 140\sqrt{2}x + 98 \quad \Rightarrow 200 x^2 + 140\sqrt{2}x -2 = 0$

Solving the above quadratic equation, we get: $x = \sin {\phi} = 0.0100005$ or $99.99499937$ . Since $\phi$ is expected to be small and therefore $\sin {\phi}$ is also small. Therefore, $\sin{\phi} = 0.0100005$ and since $\phi$ is small, $\phi \approx sin{\phi} = 0.0100005$

$\Rightarrow \theta \approx \frac {\pi}{4} - 0.0100005\ = 0.775397663$ radian $= \boxed{44.4^\circ}$ .

We know that the values of sine increases from 0 to 150

Since sin 0=0, sin 30=1/2 & sin 45=1/2^1/2

and the value 7/10 > 0 & 1/2 and <1/2^1/2

therefore the angle must be between 30 and 45 i.e 44.4 degrees