If `theta+phi=(pi)/(2)andsintheta=(1)/(2)` , then the value of `sinphi` is

Options

A. 1
B. `(1)/(sqrt(2))`
C. `(1)/(2)`
D. `(sqrt(3))/(2)`

Answer

To solve the problem step by step, we will use the given information and some trigonometric identities.

### Step 1: Understand the given equations
We are given two equations:
1. \( \theta + \phi = \frac{\pi}{2} \)
2. \( \sin \theta = \frac{1}{2} \)

### Step 2: Find the value of \( \theta \)
From the second equation, we know that \( \sin \theta = \frac{1}{2} \). The angle whose sine is \( \frac{1}{2} \) is \( \theta = \frac{\pi}{6} \) (or 30 degrees).

### Step 3: Substitute \( \theta \) into the first equation
Now, we substitute \( \theta \) into the first equation:
\[ \frac{\pi}{6} + \phi = \frac{\pi}{2} \]

### Step 4: Solve for \( \phi \)
To find \( \phi \), we rearrange the equation:
\[ \phi = \frac{\pi}{2} - \frac{\pi}{6} \]

### Step 5: Find a common denominator and simplify
To subtract these fractions, we need a common denominator. The common denominator of 2 and 6 is 6:
\[ \phi = \frac{3\pi}{6} - \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3} \]

### Step 6: Find \( \sin \phi \)
Now, we need to find \( \sin \phi \):
\[ \sin \phi = \sin \left( \frac{\pi}{3} \right) \]
From trigonometric values, we know:
\[ \sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2} \]

### Final Answer
Thus, the value of \( \sin \phi \) is:
\[ \sin \phi = \frac{\sqrt{3}}{2} \]

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