Without using tables, give the value of each of the following :
`"cosec "675^(@)`

Answer

To find the value of \( \csc 675^\circ \) without using tables, we can follow these steps:

### Step 1: Rewrite the angle
First, we can express \( 675^\circ \) in a more manageable form. We can subtract \( 720^\circ \) (which is \( 360^\circ \times 2 \)) from \( 675^\circ \):
\[ 675^\circ = 720^\circ - 45^\circ \]

### Step 2: Use the periodic property of cosecant
The cosecant function has the property that:
\[ \csc(360^\circ k - \theta) = -\csc(\theta) \]
where \( k \) is an integer. Here, \( k = 2 \) and \( \theta = 45^\circ \):
\[ \csc(675^\circ) = \csc(720^\circ - 45^\circ) = -\csc(45^\circ) \]

### Step 3: Find the value of \( \csc(45^\circ) \)
We know that:
\[ \csc(45^\circ) = \frac{1}{\sin(45^\circ)} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \]

### Step 4: Substitute back to find \( \csc(675^\circ) \)
Now we can substitute the value of \( \csc(45^\circ) \):
\[ \csc(675^\circ) = -\csc(45^\circ) = -\sqrt{2} \]

### Final Answer
Thus, the value of \( \csc(675^\circ) \) is:
\[ \csc(675^\circ) = -\sqrt{2} \]

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