The sides of the triangle are sin`alpha` , cos`alpha` and `sqrt(1+sinalphacosalpha)` for some `0 lt alpha lt (pi)/2`. Then the greatest angle of the triangle is

Options

A. `(pi)/(3)`
B. `(pi)/(2)`
C. `(2pi)/(3)`
D. `(5pi)/(6)`

Answer

To find the greatest angle of the triangle with sides \( a = \sin \alpha \), \( b = \cos \alpha \), and \( c = \sqrt{1 + \sin \alpha \cos \alpha} \), we can follow these steps:

### Step 1: Identify the sides of the triangle
Let:
- \( a = \sin \alpha \)
- \( b = \cos \alpha \)
- \( c = \sqrt{1 + \sin \alpha \cos \alpha} \)

### Step 2: Use the cosine rule to find the angle opposite to side \( c \)
The cosine rule states that for any triangle with sides \( a, b, c \) opposite to angles \( A, B, C \) respectively:

\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]

### Step 3: Substitute the values of \( a, b, c \) into the cosine rule
Substituting \( a, b, c \):

\[ \cos C = \frac{(\sin \alpha)^2 + (\cos \alpha)^2 - (\sqrt{1 + \sin \alpha \cos \alpha})^2}{2(\sin \alpha)(\cos \alpha)} \]

### Step 4: Simplify the expression
Using the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \):

\[ \cos C = \frac{1 - (1 + \sin \alpha \cos \alpha)}{2 \sin \alpha \cos \alpha} \]

This simplifies to:

\[ \cos C = \frac{1 - 1 - \sin \alpha \cos \alpha}{2 \sin \alpha \cos \alpha} = \frac{-\sin \alpha \cos \alpha}{2 \sin \alpha \cos \alpha} \]

### Step 5: Further simplification
This gives:

\[ \cos C = -\frac{1}{2} \]

### Step 6: Find the angle \( C \)
The angle \( C \) corresponding to \( \cos C = -\frac{1}{2} \) is:

\[ C = 120^\circ \]

### Step 7: Determine the greatest angle of the triangle
Since \( C \) is the angle opposite the longest side \( c \), it is the greatest angle of the triangle.

Thus, the greatest angle of the triangle is:

\[ \boxed{120^\circ} \]

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