What is the acute angle $A$ if $\log_4(\sin^2A)=-1$?

Question

If angle $A$ is acute and $\log_4(\sin^2A)=-1$, then the value of $A$, to the nearest tenth of a radian, is...

I've worked through this some different ways but I haven't made any progress. Please give me a hint as to how to proceed.

Answer 1

Hint:

Unchain the various operations from each other:

$$\log_4 q = -1\\ r^2 = q\\ \sin A = r$$

and finally, $A$ is an acute angle.

Different direction hint:

Taking the rule $\log a^b = b\log a$, we can start with $\log_4(\sin^2 A) = -1 = 2\log_4(\sin A)$.

Answer 2

We want to find the value of $A$ in the equation $\log_4{\sin^2A}=-1$.

Note that if $y=\log_a{x}$ this means that $a^y=x$.

Using the definition above we have $4^{-1}=\sin^2A$ which in turn means $\frac{1}{4}=\sin^2A$.

Then $\pm\frac{1}{2}=\sin A$. Finally we have $A=\sin^{-1}\frac{1}{2}$ and $A=\sin^{-1}-\frac{1}{2}$. So the answer is $\frac{\pi}{6}$ and its negative counter part.