If $ heta$ and $\phi$ are angles in the first quadrant such that $tan~ heta=\frac{1}{7}$ and $sin~\phi=\frac{1}{\sqrt{10}}$ then which of the following is correct?

Q: If $\theta$ and $\phi$ are angles in the first quadrant such that $tan~\theta=\frac{1}{7}$ and $sin~\phi=\frac{1}{\sqrt{10}}$ then which of the following is correct?

If $\theta$ and $\phi$ are angles in the first quadrant such that $tan~\theta=\frac{1}{7}$ and $sin~\phi=\frac{1}{\sqrt{10}}$ then which of the following is correct?

Question

If $	heta$ and $\phi$ are angles in the first quadrant such that $tan~	heta=\frac{1}{7}$ and $sin~\phi=\frac{1}{\sqrt{10}}$ then which of the following is correct?

Accepted Answer

To determine why Option 2 is the correct answer, we need to analyze the given conditions and perform some trigonometric calculations.

### Given:
1. $ \tan \theta = \frac{1}{7} $
2. $ \sin \phi = \frac{1}{\sqrt{10}} $

### Objective:
Find $ \theta + 2\phi $.

### Explanation:

#### Step 1: Determine $\theta$

Since $\tan \theta = \frac{1}{7}$, we can use the identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$. We know:

$$
\tan \theta = \frac{1}{7} \implies \sin \theta = \frac{1}{\sqrt{1 + \left(\frac{1}{7}\right)^2}} = \frac{1}{\sqrt{1 + \frac{1}{49}}} = \frac{1}{\sqrt{\frac{50}{49}}} = \frac{7}{\sqrt{50}}
$$

$$
\cos \theta = \frac{1}{7} \times \frac{7}{\sqrt{50}} = \frac{1}{\sqrt{50}}
$$

#### Step 2: Determine $\phi$

Given $\sin \phi = \frac{1}{\sqrt{10}}$, we can find $\cos \phi$ using the Pythagorean identity:

$$
\cos \phi = \sqrt{1 - \sin^2 \phi} = \sqrt{1 - \left(\frac{1}{\sqrt{10}}\right)^2} = \sqrt{1 - \frac{1}{10}} = \sqrt{\frac{9}{10}} = \frac{3}{\sqrt{10}}
$$

#### Step 3: Calculate $\theta + 2\phi$

To find $2\phi$, we use the double angle formula for sine:

$$
\sin 2\phi = 2 \sin \phi \cos \phi = 2 \times \frac{1}{\sqrt{10}} \times \frac{3}{\sqrt{10}} = \frac{6}{10} = 0.6
$$

Now, we need to find $\theta + 2\phi$. Since $\theta$ and $2\phi$ are both in the first quadrant, we can approximate their sum using the known values:

- $\theta \approx \tan^{-1}\left(\frac{1}{7}\right)$
- $2\phi \approx \sin^{-1}(0.6)$

Using approximate values:
- $\theta \approx 8.13^\circ$ (since $\tan^{-1}\left(\frac{1}{7}\right) \approx 8.13^\circ$)
- $2\phi \approx 36.87^\circ$ (since $\sin^{-1}(0.6) \approx 36.87^\circ$)

Thus, $\theta + 2\phi \approx 8.13^\circ + 36.87^\circ = 45^\circ$.

### Conclusion:

- **Option 2: $\theta + 2\phi = 45^\circ$** is correct because the calculated sum of $\theta$ and $2\phi$ is approximately $45^\circ$.

### Why Other Options are Incorrect:

1. **Option 1: $\theta + 2\phi = 30^\circ$** is incorrect because the calculated sum is greater than $30^\circ$.
2. **Option 3: $\theta + 2\phi = 75^\circ$** is incorrect because the calculated sum is less than $75^\circ$.
3. **Option 4: $\theta + 2\phi = 90^\circ$** is incorrect because the calculated sum is less than $90^\circ$.

Therefore, the correct answer is indeed Option 2, $\theta + 2\phi = 45^\circ$.