If `sin (30 ^(@) - theta) = cos (60 ^(@) + phi)` , then

Options

A. `phi - theta = 30^(@)`
B. ` phi - theta = 0^(@)`
C. ` phi + theta = 60^(@)`
D. ` phi - theta = 60^(@)`

Answer

To solve the equation \( \sin(30^\circ - \theta) = \cos(60^\circ + \phi) \), we can follow these steps:

### Step 1: Use Trigonometric Identities
We know that \( \cos(60^\circ + \phi) \) can be rewritten using the co-function identity:
\[ \cos(60^\circ + \phi) = \sin(90^\circ - (60^\circ + \phi)) = \sin(30^\circ - \phi) \]

### Step 2: Set the Equations Equal
Now we can rewrite our original equation:
\[ \sin(30^\circ - \theta) = \sin(30^\circ - \phi) \]

### Step 3: Apply the Sine Identity
From the property of sine, if \( \sin A = \sin B \), then:
\[ A = B + n \cdot 180^\circ \quad \text{or} \quad A = 180^\circ - B + n \cdot 180^\circ \quad (n \in \mathbb{Z}) \]
Here, we can set:
1. \( 30^\circ - \theta = 30^\circ - \phi \)
2. \( 30^\circ - \theta = 180^\circ - (30^\circ - \phi) \)

### Step 4: Solve the First Equation
From the first equation:
\[ 30^\circ - \theta = 30^\circ - \phi \]
Subtracting \( 30^\circ \) from both sides gives:
\[ -\theta = -\phi \quad \Rightarrow \quad \theta = \phi \]

### Step 5: Solve the Second Equation
From the second equation:
\[ 30^\circ - \theta = 180^\circ - 30^\circ + \phi \]
This simplifies to:
\[ 30^\circ - \theta = 150^\circ + \phi \]
Rearranging gives:
\[ -\theta - \phi = 150^\circ - 30^\circ \quad \Rightarrow \quad -\theta - \phi = 120^\circ \]
Thus:
\[ \theta + \phi = -120^\circ \]

### Conclusion
From the first equation, we found \( \theta = \phi \). Therefore, we can conclude that:
\[ \phi - \theta = 0 \]

### Final Answer
The correct answer is \( \phi - \theta = 0 \).

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