Biggest among
`(sin1+cos1),(sqrt(sin1))+sqrt(cos1)),sqrt(1+sin2)and 1, is`

Options

A. `sin1 +cos1`
B. `sqrt(sin1)+sqrt(cos1)`
C. `sqrt(1+sin2)`
D. 1

Explanation

To find the greatest value among the four expressions: \( ( \sin 1 + \cos 1 ) \), \( \sqrt{\sin 1} + \sqrt{\cos 1} \), \( \sqrt{1 + \sin 2} \), and \( 1 \), we will analyze each expression step by step.

### Step 1: Evaluate \( \sin 1 + \cos 1 \)

Using the known values of sine and cosine:
- \( \sin 1 \approx 0.8415 \)
- \( \cos 1 \approx 0.5403 \)

Calculating:
\[ \sin 1 + \cos 1 \approx 0.8415 + 0.5403 \approx 1.3818 \]

### Step 2: Evaluate \( \sqrt{\sin 1} + \sqrt{\cos 1} \)

Calculating the square roots:
\[ \sqrt{\sin 1} \approx \sqrt{0.8415} \approx 0.917 \]
\[ \sqrt{\cos 1} \approx \sqrt{0.5403} \approx 0.735 \]
Adding these values:
\[ \sqrt{\sin 1} + \sqrt{\cos 1} \approx 0.917 + 0.735 \approx 1.652 \]

### Step 3: Evaluate \( \sqrt{1 + \sin 2} \)

Using the double angle formula:
\[ \sin 2 = 2 \sin 1 \cos 1 \approx 2 \times 0.8415 \times 0.5403 \approx 0.9093 \]
Calculating:
\[ 1 + \sin 2 \approx 1 + 0.9093 \approx 1.9093 \]
Now taking the square root:
\[ \sqrt{1 + \sin 2} \approx \sqrt{1.9093} \approx 1.38 \]

### Step 4: Compare all values

Now we have:
1. \( \sin 1 + \cos 1 \approx 1.3818 \)
2. \( \sqrt{\sin 1} + \sqrt{\cos 1} \approx 1.652 \)
3. \( \sqrt{1 + \sin 2} \approx 1.38 \)
4. \( 1 \)

### Conclusion

Comparing the values:
- \( 1.3818 \) (from \( \sin 1 + \cos 1 \))
- \( 1.652 \) (from \( \sqrt{\sin 1} + \sqrt{\cos 1} \))
- \( 1.38 \) (from \( \sqrt{1 + \sin 2} \))
- \( 1 \)

The greatest value among these is:
\[ \sqrt{\sin 1} + \sqrt{\cos 1} \approx 1.652 \]

Thus, the biggest among the given expressions is:
\[ \boxed{\sqrt{\sin 1} + \sqrt{\cos 1}} \]