Without using tables, give the value of each of the following :
`cos855^(@)`

Answer

To find the value of \( \cos 855^\circ \) without using tables, we can follow these steps:

### Step 1: Rewrite the angle
First, we can express \( 855^\circ \) in a more manageable form. We can write:
\[ 855^\circ = 900^\circ - 45^\circ \]

### Step 2: Use the cosine subtraction formula
Now, we can use the cosine subtraction formula, which states:
\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]
In our case, \( a = 900^\circ \) and \( b = 45^\circ \):
\[ \cos(855^\circ) = \cos(900^\circ - 45^\circ) \]

### Step 3: Calculate \( \cos 900^\circ \) and \( \sin 900^\circ \)
From the unit circle, we know:
\[ \cos 900^\circ = 0 \quad \text{and} \quad \sin 900^\circ = 1 \]

### Step 4: Substitute values into the formula
Substituting these values into the cosine subtraction formula gives:
\[ \cos(855^\circ) = \cos(900^\circ) \cos(45^\circ) + \sin(900^\circ) \sin(45^\circ) \]
\[ = 0 \cdot \cos(45^\circ) + 1 \cdot \sin(45^\circ) \]
\[ = 0 + \sin(45^\circ) \]

### Step 5: Calculate \( \sin 45^\circ \)
We know that:
\[ \sin(45^\circ) = \frac{1}{\sqrt{2}} \]

### Step 6: Final result
Thus, we have:
\[ \cos(855^\circ) = \frac{1}{\sqrt{2}} \]

### Step 7: Consider the quadrant
Since \( 855^\circ \) is in the fourth quadrant (as it is between \( 720^\circ \) and \( 900^\circ \)), the cosine value will be positive. Therefore, we conclude:
\[ \cos(855^\circ) = \frac{1}{\sqrt{2}} \]

### Summary
The value of \( \cos 855^\circ \) is \( \frac{1}{\sqrt{2}} \).

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