Find the values of:
(i) `tan (-30^(@))` (ii) `sin 120^(@)` (iii) `sin 135^(@)` (iv) `cos 150^(@)` (v) `sin 270^(@)` (vi) `cos 270^(@)`

Options

A. `120^(@)`
B. `pi/3 rad`
C. `(2pi)/3 rad`
D. `160^(@)`

Explanation

To find the values of the trigonometric functions given in the question, we will solve each part step by step.

### Solution:

**(i) Find `tan(-30°`:**

Using the property of tangent:
\[ \tan(-\theta) = -\tan(\theta) \]
So,
\[ \tan(-30°) = -\tan(30°) \]
We know that:
\[ \tan(30°) = \frac{1}{\sqrt{3}} \]
Thus,
\[ \tan(-30°) = -\frac{1}{\sqrt{3}} \]

**(ii) Find `sin(120°)`:**

We can express \(120°\) as:
\[ 120° = 90° + 30° \]
Using the sine addition formula:
\[ \sin(90° + \theta) = \cos(\theta) \]
So,
\[ \sin(120°) = \cos(30°) \]
And we know:
\[ \cos(30°) = \frac{\sqrt{3}}{2} \]
Thus,
\[ \sin(120°) = \frac{\sqrt{3}}{2} \]

**(iii) Find `sin(135°)`:**

We can express \(135°\) as:
\[ 135° = 90° + 45° \]
Using the sine addition formula:
\[ \sin(90° + \theta) = \cos(\theta) \]
So,
\[ \sin(135°) = \cos(45°) \]
And we know:
\[ \cos(45°) = \frac{1}{\sqrt{2}} \]
Thus,
\[ \sin(135°) = \frac{1}{\sqrt{2}} \]

**(iv) Find `cos(150°)`:**

We can express \(150°\) as:
\[ 150° = 180° - 30° \]
Using the cosine subtraction formula:
\[ \cos(180° - \theta) = -\cos(\theta) \]
So,
\[ \cos(150°) = -\cos(30°) \]
And we know:
\[ \cos(30°) = \frac{\sqrt{3}}{2} \]
Thus,
\[ \cos(150°) = -\frac{\sqrt{3}}{2} \]

**(v) Find `sin(270°)`:**

We can express \(270°\) as:
\[ 270° = 180° + 90° \]
Using the sine addition formula:
\[ \sin(180° + \theta) = -\sin(\theta) \]
So,
\[ \sin(270°) = -\sin(90°) \]
And we know:
\[ \sin(90°) = 1 \]
Thus,
\[ \sin(270°) = -1 \]

**(vi) Find `cos(270°)`:**

We can express \(270°\) as:
\[ 270° = 180° + 90° \]
Using the cosine addition formula:
\[ \cos(180° + \theta) = -\cos(\theta) \]
So,
\[ \cos(270°) = -\cos(90°) \]
And we know:
\[ \cos(90°) = 0 \]
Thus,
\[ \cos(270°) = 0 \]

### Final Answers:
1. \( \tan(-30°) = -\frac{1}{\sqrt{3}} \)
2. \( \sin(120°) = \frac{\sqrt{3}}{2} \)
3. \( \sin(135°) = \frac{1}{\sqrt{2}} \)
4. \( \cos(150°) = -\frac{\sqrt{3}}{2} \)
5. \( \sin(270°) = -1 \)
6. \( \cos(270°) = 0 \)