The y- intercept of y=`|sqrt2csc(x+(pi)/(5))|` is

Options

A. 0.22
B. 0.67
C. 1.41
D. 2.4

Explanation

To find the y-intercept of the function \( y = |\sqrt{2} \csc\left(x + \frac{\pi}{5}\right)| \), we will follow these steps:

### Step 1: Understand the y-intercept
The y-intercept of a function is the value of \( y \) when \( x = 0 \).

### Step 2: Substitute \( x = 0 \) into the equation
We substitute \( x = 0 \) into the equation:
\[ y = |\sqrt{2} \csc\left(0 + \frac{\pi}{5}\right)| \]
This simplifies to:
\[ y = |\sqrt{2} \csc\left(\frac{\pi}{5}\right)| \]

### Step 3: Use the definition of cosecant
Recall that cosecant is the reciprocal of sine:
\[ \csc\left(\frac{\pi}{5}\right) = \frac{1}{\sin\left(\frac{\pi}{5}\right)} \]
So, we can rewrite \( y \) as:
\[ y = |\sqrt{2} \cdot \frac{1}{\sin\left(\frac{\pi}{5}\right)}| \]

### Step 4: Calculate \( \sin\left(\frac{\pi}{5}\right) \)
Using a calculator or trigonometric tables, we find:
\[ \sin\left(\frac{\pi}{5}\right) \approx 0.5878 \]

### Step 5: Substitute the sine value back into the equation
Now, substitute this value back into the equation:
\[ y = |\sqrt{2} \cdot \frac{1}{0.5878}| \]

### Step 6: Calculate \( \sqrt{2} \)
We know that:
\[ \sqrt{2} \approx 1.4142 \]

### Step 7: Calculate \( y \)
Now, we can calculate \( y \):
\[ y \approx |1.4142 \cdot \frac{1}{0.5878}| \]
Calculating this gives:
\[ y \approx |1.4142 \cdot 1.6990| \approx |2.3980| \]

### Step 8: Final result
Since \( y \) is inside absolute value, we have:
\[ y \approx 2.3980 \]
Thus, the y-intercept is approximately \( 2.4 \).

### Summary
The y-intercept of the function \( y = |\sqrt{2} \csc\left(x + \frac{\pi}{5}\right)| \) is approximately \( 2.4 \).

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