Find the values of the following :-
`{:((a)sin(-30^(@)),(b)cos(-45^(@))),((c)sin(-60^(@)),(d)tan(-45^(@))):}`

Answer

To solve the given question, we need to find the values of the trigonometric functions for the specified angles. Let's break it down step by step.

### Step 1: Find \( \sin(-30^\circ) \)
Using the property of sine, we know that:
\[ \sin(-\theta) = -\sin(\theta) \]
Thus,
\[ \sin(-30^\circ) = -\sin(30^\circ) \]
From trigonometric tables, we know:
\[ \sin(30^\circ) = \frac{1}{2} \]
Therefore,
\[ \sin(-30^\circ) = -\frac{1}{2} \]

### Step 2: Find \( \cos(-45^\circ) \)
Using the property of cosine, we know that:
\[ \cos(-\theta) = \cos(\theta) \]
Thus,
\[ \cos(-45^\circ) = \cos(45^\circ) \]
From trigonometric tables, we know:
\[ \cos(45^\circ) = \frac{1}{\sqrt{2}} \]
Therefore,
\[ \cos(-45^\circ) = \frac{1}{\sqrt{2}} \]

### Step 3: Find \( \sin(-60^\circ) \)
Using the property of sine again:
\[ \sin(-\theta) = -\sin(\theta) \]
Thus,
\[ \sin(-60^\circ) = -\sin(60^\circ) \]
From trigonometric tables, we know:
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]
Therefore,
\[ \sin(-60^\circ) = -\frac{\sqrt{3}}{2} \]

### Step 4: Find \( \tan(-45^\circ) \)
Using the property of tangent, we know that:
\[ \tan(-\theta) = -\tan(\theta) \]
Thus,
\[ \tan(-45^\circ) = -\tan(45^\circ) \]
From trigonometric tables, we know:
\[ \tan(45^\circ) = 1 \]
Therefore,
\[ \tan(-45^\circ) = -1 \]

### Final Answers
Now, we can summarize the answers:
- \( \sin(-30^\circ) = -\frac{1}{2} \)
- \( \cos(-45^\circ) = \frac{1}{\sqrt{2}} \)
- \( \sin(-60^\circ) = -\frac{\sqrt{3}}{2} \)
- \( \tan(-45^\circ) = -1 \)