If : `sin (theta - 30^(@))+cos (theta - 60^(@))=k * sin theta, "then" : k=`

Options

A. `1//sqrt3`
B. `-1//sqrt3`
C. `sqrt3`
D. `-sqrt3`

Answer

To solve the equation \( \sin(\theta - 30^\circ) + \cos(\theta - 60^\circ) = k \sin \theta \), we will expand both sides using trigonometric identities and then compare coefficients to find the value of \( k \).

### Step 1: Expand \( \sin(\theta - 30^\circ) \)
Using the sine subtraction formula:
\[ \sin(a - b) = \sin a \cos b - \cos a \sin b \]
we can expand \( \sin(\theta - 30^\circ) \):
\[ \sin(\theta - 30^\circ) = \sin \theta \cos 30^\circ - \cos \theta \sin 30^\circ \]
Substituting the values of \( \cos 30^\circ = \frac{\sqrt{3}}{2} \) and \( \sin 30^\circ = \frac{1}{2} \):
\[ \sin(\theta - 30^\circ) = \sin \theta \cdot \frac{\sqrt{3}}{2} - \cos \theta \cdot \frac{1}{2} \]

### Step 2: Expand \( \cos(\theta - 60^\circ) \)
Using the cosine subtraction formula:
\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]
we can expand \( \cos(\theta - 60^\circ) \):
\[ \cos(\theta - 60^\circ) = \cos \theta \cos 60^\circ + \sin \theta \sin 60^\circ \]
Substituting the values of \( \cos 60^\circ = \frac{1}{2} \) and \( \sin 60^\circ = \frac{\sqrt{3}}{2} \):
\[ \cos(\theta - 60^\circ) = \cos \theta \cdot \frac{1}{2} + \sin \theta \cdot \frac{\sqrt{3}}{2} \]

### Step 3: Combine the expansions
Now we combine both expansions:
\[ \sin(\theta - 30^\circ) + \cos(\theta - 60^\circ) = \left(\sin \theta \cdot \frac{\sqrt{3}}{2} - \cos \theta \cdot \frac{1}{2}\right) + \left(\cos \theta \cdot \frac{1}{2} + \sin \theta \cdot \frac{\sqrt{3}}{2}\right) \]
Notice that the \( -\cos \theta \cdot \frac{1}{2} \) and \( +\cos \theta \cdot \frac{1}{2} \) cancel each other out:
\[ = \sin \theta \cdot \frac{\sqrt{3}}{2} + \sin \theta \cdot \frac{\sqrt{3}}{2} = 2 \cdot \sin \theta \cdot \frac{\sqrt{3}}{2} \]
This simplifies to:
\[ = \sqrt{3} \sin \theta \]

### Step 4: Set the equation equal to \( k \sin \theta \)
We have:
\[ \sqrt{3} \sin \theta = k \sin \theta \]
Assuming \( \sin \theta \neq 0 \), we can divide both sides by \( \sin \theta \):
\[ k = \sqrt{3} \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{\sqrt{3}} \]