Without using tables, give the value of the following :
`"sec"(15pi)/(4)`

Answer

To find the value of \( \sec\left(\frac{15\pi}{4}\right) \) without using tables, we can follow these steps:

### Step 1: Simplify the angle
We start by simplifying the angle \( \frac{15\pi}{4} \). We can express it in terms of a more manageable angle by subtracting multiples of \( 2\pi \) (which is \( \frac{8\pi}{4} \)):

\[ \frac{15\pi}{4} = \frac{16\pi}{4} - \frac{\pi}{4} = 4\pi - \frac{\pi}{4} \]

### Step 2: Determine the equivalent angle in standard position
The angle \( 4\pi - \frac{\pi}{4} \) is equivalent to \( -\frac{\pi}{4} \) when considering the periodicity of the trigonometric functions. Since \( \sec(x) \) has a period of \( 2\pi \), we can also represent this as:

\[ \sec\left(4\pi - \frac{\pi}{4}\right) = \sec\left(-\frac{\pi}{4}\right) \]

### Step 3: Identify the quadrant
The angle \( -\frac{\pi}{4} \) lies in the fourth quadrant. In the fourth quadrant, the cosine function (and therefore secant) is positive.

### Step 4: Use the reference angle
The reference angle for \( -\frac{\pi}{4} \) is \( \frac{\pi}{4} \). Thus, we can write:

\[ \sec\left(-\frac{\pi}{4}\right) = \sec\left(\frac{\pi}{4}\right) \]

### Step 5: Calculate the secant
Now, we need to find \( \sec\left(\frac{\pi}{4}\right) \). We know that:

\[ \sec(x) = \frac{1}{\cos(x)} \]

For \( x = \frac{\pi}{4} \):

\[ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

Thus,

\[ \sec\left(\frac{\pi}{4}\right) = \frac{1}{\cos\left(\frac{\pi}{4}\right)} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]

### Final Result
Therefore, the value of \( \sec\left(\frac{15\pi}{4}\right) \) is:

\[ \sec\left(\frac{15\pi}{4}\right) = \sqrt{2} \]