### Instruction: Given that $\theta$ and $\phi$ are acute angles such that $\tan \theta = \frac{1}{7}$ and $\sin \phi = \frac{1}{\sqrt{10}},$ find $\theta + 2 \phi,$ measured in radians.### Response: I want to find the sum of $\theta$ and $2 \phi,$ but I only have information about their trigonometric ratios.
I wonder if there is a way to relate the angles to their ratios using some identity or formula.
I recall that there is an inverse tangent function, denoted by $\tan^{-1},$ that gives the angle whose tangent is a given value.
For example, $\tan^{-1} \frac{1}{7}$ means the angle whose tangent is $\frac{1}{7}.$
Therefore, I can use the inverse tangent function to find $\theta$ from $\tan \theta = \frac{1}{7}.$
Similarly, I can use the inverse sine function, denoted by $\sin^{-1},$ to find $\phi$ from $\sin \phi = \frac{1}{\sqrt{10}}.$
I can use a calculator to evaluate these inverse functions, but I need to make sure they are in radian mode, not degree mode.
Using a calculator, I find that $\tan^{-1} \frac{1}{7} \approx 0.1419$ radians and $\sin^{-1} \frac{1}{\sqrt{10}} \approx 0.3218$ radians.
Now I can plug these values into the expression $\theta + 2 \phi$ and simplify.
$\theta + 2 \phi \approx 0.1419 + 2(0.3218) \approx 0.7855$ radians.