Without using tables, give the value of each of the following :
`sec210^(@)`

Answer

To find the value of \( \sec 210^\circ \) without using tables, we can follow these steps:

### Step 1: Identify the angle in terms of a known angle
We can express \( 210^\circ \) as:
\[ 210^\circ = 180^\circ + 30^\circ \]

### Step 2: Determine the quadrant
Since \( 210^\circ \) is in the third quadrant, we know that the secant function (which is the reciprocal of the cosine function) will be negative in this quadrant.

### Step 3: Use the identity for secant
We can use the identity:
\[ \sec(180^\circ + \theta) = -\sec(\theta) \]
In our case, \( \theta = 30^\circ \):
\[ \sec(210^\circ) = \sec(180^\circ + 30^\circ) = -\sec(30^\circ) \]

### Step 4: Find the value of \( \sec(30^\circ) \)
The value of \( \sec(30^\circ) \) is the reciprocal of \( \cos(30^\circ) \):
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \implies \sec(30^\circ) = \frac{1}{\cos(30^\circ)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]

### Step 5: Substitute back to find \( \sec(210^\circ) \)
Now substituting back into our equation:
\[ \sec(210^\circ) = -\sec(30^\circ) = -\frac{2}{\sqrt{3}} \]

### Final Answer
Thus, the value of \( \sec 210^\circ \) is:
\[ \sec 210^\circ = -\frac{2}{\sqrt{3}} \]

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