Find the values of the following allied angles :
(iii) `tan 210^(@)`

Answer

To find the value of \( \tan 210^\circ \), we can follow these steps:

### Step 1: Identify the Quadrant
The angle \( 210^\circ \) is located in the third quadrant of the unit circle. Angles in the third quadrant range from \( 180^\circ \) to \( 270^\circ \).

**Hint:** Remember that angles are measured from the positive x-axis, and the third quadrant is where both sine and cosine values are negative.

### Step 2: Express the Angle in Terms of Reference Angle
The reference angle for \( 210^\circ \) can be found by subtracting \( 180^\circ \):
\[ 210^\circ - 180^\circ = 30^\circ \]
So, the reference angle is \( 30^\circ \).

**Hint:** The reference angle is the acute angle formed by the terminal side of the given angle and the x-axis.

### Step 3: Use the Tangent Function in the Third Quadrant
In the third quadrant, the tangent function is positive. Therefore, we can express \( \tan 210^\circ \) as:
\[ \tan 210^\circ = \tan(180^\circ + 30^\circ) \]
Using the tangent addition formula, we have:
\[ \tan(180^\circ + \theta) = \tan \theta \]
Thus,
\[ \tan 210^\circ = \tan 30^\circ \]

**Hint:** Recall that in the third quadrant, tangent is positive, which is why we can use the reference angle directly.

### Step 4: Find the Value of \( \tan 30^\circ \)
The value of \( \tan 30^\circ \) is known:
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \]

**Hint:** Memorizing the values of tangent for common angles (like \( 30^\circ, 45^\circ, \) and \( 60^\circ \)) can help you solve problems quickly.

### Step 5: Conclusion
Putting it all together, we find:
\[ \tan 210^\circ = \tan 30^\circ = \frac{1}{\sqrt{3}} \]

Thus, the value of \( \tan 210^\circ \) is:
\[ \boxed{\frac{1}{\sqrt{3}}} \]