If `sinalpha=1/sqrt5 and sinbeta=3/5` , then `beta-alpha` lies in

Options

A. `[0,pi//4]`
B. `[pi//2,3pi//4]`
C. `[3pi//4,pi]`
D. none of these

Explanation

To solve the problem, we will follow these steps:

### Step 1: Find cos(α)
Given that \( \sin \alpha = \frac{1}{\sqrt{5}} \), we can use the Pythagorean identity:

\[ \sin^2 \alpha + \cos^2 \alpha = 1 \]

Substituting the value of \( \sin \alpha \):

\[ \left(\frac{1}{\sqrt{5}}\right)^2 + \cos^2 \alpha = 1 \]

This simplifies to:

\[ \frac{1}{5} + \cos^2 \alpha = 1 \]

Now, solving for \( \cos^2 \alpha \):

\[ \cos^2 \alpha = 1 - \frac{1}{5} = \frac{4}{5} \]

Taking the square root gives:

\[ \cos \alpha = \frac{2}{\sqrt{5}} \quad \text{(since α is in the first quadrant)} \]

### Step 2: Find cos(β)
Given that \( \sin \beta = \frac{3}{5} \), we again use the Pythagorean identity:

\[ \sin^2 \beta + \cos^2 \beta = 1 \]

Substituting the value of \( \sin \beta \):

\[ \left(\frac{3}{5}\right)^2 + \cos^2 \beta = 1 \]

This simplifies to:

\[ \frac{9}{25} + \cos^2 \beta = 1 \]

Now, solving for \( \cos^2 \beta \):

\[ \cos^2 \beta = 1 - \frac{9}{25} = \frac{16}{25} \]

Taking the square root gives:

\[ \cos \beta = \frac{4}{5} \quad \text{(since β is in the first quadrant)} \]

### Step 3: Calculate sin(β - α)
Using the sine difference formula:

\[ \sin(\beta - \alpha) = \sin \beta \cos \alpha - \cos \beta \sin \alpha \]

Substituting the values we found:

\[ \sin(\beta - \alpha) = \left(\frac{3}{5}\right) \left(\frac{2}{\sqrt{5}}\right) - \left(\frac{4}{5}\right) \left(\frac{1}{\sqrt{5}}\right) \]

Calculating this gives:

\[ \sin(\beta - \alpha) = \frac{6}{5\sqrt{5}} - \frac{4}{5\sqrt{5}} = \frac{2}{5\sqrt{5}} \]

### Step 4: Determine the range of β - α
To find the range of \( \beta - \alpha \), we need to evaluate \( \sin(\beta - \alpha) \):

\[ \sin(\beta - \alpha) = \frac{2}{5\sqrt{5}} \approx 0.178 \]

Since \( \sin x \) is increasing in the interval \( [0, \frac{\pi}{2}] \), we can compare \( \sin(\beta - \alpha) \) with known sine values:

- \( \sin(0) = 0 \)
- \( \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \approx 0.707 \)

Since \( 0 < 0.178 < 0.707 \), we conclude:

\[ 0 < \beta - \alpha < \frac{\pi}{4} \]

### Final Answer
Thus, \( \beta - \alpha \) lies in the interval \( (0, \frac{\pi}{4}) \).

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