Which among the following is an irrational quantity ?
(a) `tan 30^(@) tan 60^(@)`
(b) `sin 30 ^(@)`
(c ) `tan 45 ^(@)`
(d) `cos 30^(@)`

Options

A. (d)
B. (a)
C. (b)
D. (c)

Answer

To determine which of the given quantities is irrational, we will evaluate each option step by step.

### Step 1: Evaluate \( \tan 30^\circ \) and \( \tan 60^\circ \)

- The value of \( \tan 30^\circ \) is \( \frac{1}{\sqrt{3}} \) or \( \frac{\sqrt{3}}{3} \).
- The value of \( \tan 60^\circ \) is \( \sqrt{3} \).

Now, we multiply these two values:

\[ \tan 30^\circ \cdot \tan 60^\circ = \left(\frac{\sqrt{3}}{3}\right) \cdot \sqrt{3} = \frac{3}{3} = 1 \]

### Step 2: Evaluate \( \sin 30^\circ \)

- The value of \( \sin 30^\circ \) is \( \frac{1}{2} \).

### Step 3: Evaluate \( \tan 45^\circ \)

- The value of \( \tan 45^\circ \) is \( 1 \).

### Step 4: Evaluate \( \cos 30^\circ \)

- The value of \( \cos 30^\circ \) is \( \frac{\sqrt{3}}{2} \).

### Summary of Evaluated Values

- \( \tan 30^\circ \cdot \tan 60^\circ = 1 \) (Rational)
- \( \sin 30^\circ = \frac{1}{2} \) (Rational)
- \( \tan 45^\circ = 1 \) (Rational)
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \) (Irrational)

### Conclusion

Among the options given, the only irrational quantity is \( \cos 30^\circ \), which is \( \frac{\sqrt{3}}{2} \).

Thus, the answer is **(d) \( \cos 30^\circ \)**.

---