### 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 see that I need to find the values of $\theta$ and $\phi$ in radians, and then add them together with a factor of 2 on $\phi$.
I also notice that I have the tangent and sine ratios for the two angles, which suggests using right triangles and inverse trigonometric functions.
To find $\theta,$ I can draw a right triangle with $\theta$ as one of the acute angles, and label the opposite and adjacent sides as 1 and 7, respectively, based on the definition of tangent.
Then, by the Pythagorean theorem, the hypotenuse of this triangle is $\sqrt{1^2 + 7^2} = \sqrt{50}.$
Now, I can use the inverse tangent function to find $\theta,$ since $\tan \theta = \frac{1}{7}.$
This gives me $\theta = \tan^{-1} \frac{1}{7} \approx 0.1419$ radians, rounding to four decimal places.
Similarly, to find $\phi,$ I can draw another right triangle with $\phi$ as one of the acute angles, and label the opposite and hypotenuse sides as 1 and $\sqrt{10},$ respectively, based on the definition of sine.
Then, by the Pythagorean theorem, the adjacent side of this triangle is $\sqrt{\sqrt{10}^2 - 1^2} = \sqrt{9} = 3.$
Now, I can use the inverse sine function to find $\phi,$ since $\sin \phi = \frac{1}{\sqrt{10}}.$
This gives me $\phi = \sin^{-1} \frac{1}{\sqrt{10}} \approx 0.3218$ radians, rounding to four decimal places.
Finally, to find $\theta + 2 \phi,$ I just need to add the two angles and multiply $\phi$ by 2.
This gives me $\theta + 2 \phi \approx 0.1419 + 2 \times 0.3218 \approx 0.7855$ radians, rounding to four decimal places.