If ` sin theta = sin 15^(@) + sin 45^(@), " where " 0^(@) lt theta lt 90^(@),` then value of `theta` is

Options

A. `45^(@)`
B. ` 54^(@)`
C. `60^(@)`
D. `75^(@)`

Answer

To solve the problem \( \sin \theta = \sin 15^\circ + \sin 45^\circ \) where \( 0^\circ < \theta < 90^\circ \), we can follow these steps:

### Step 1: Write the equation
We start with the equation:
\[ \sin \theta = \sin 15^\circ + \sin 45^\circ \]

### Step 2: Use the sine addition formula
We can use the sine addition formula:
\[ \sin x + \sin y = 2 \sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) \]
In our case, let \( x = 15^\circ \) and \( y = 45^\circ \).

### Step 3: Calculate \( x+y \) and \( x-y \)
Calculate \( x+y \) and \( x-y \):
\[ x + y = 15^\circ + 45^\circ = 60^\circ \]
\[ x - y = 15^\circ - 45^\circ = -30^\circ \]

### Step 4: Substitute into the formula
Now substitute these values into the sine addition formula:
\[ \sin 15^\circ + \sin 45^\circ = 2 \sin\left(\frac{60^\circ}{2}\right) \cos\left(\frac{-30^\circ}{2}\right) \]
This simplifies to:
\[ \sin 15^\circ + \sin 45^\circ = 2 \sin(30^\circ) \cos(-15^\circ) \]

### Step 5: Evaluate \( \sin(30^\circ) \) and \( \cos(-15^\circ) \)
We know:
\[ \sin(30^\circ) = \frac{1}{2} \]
And since cosine is an even function:
\[ \cos(-15^\circ) = \cos(15^\circ) \]
Thus:
\[ \sin 15^\circ + \sin 45^\circ = 2 \cdot \frac{1}{2} \cdot \cos(15^\circ) = \cos(15^\circ) \]

### Step 6: Set the equation
Now we have:
\[ \sin \theta = \cos(15^\circ) \]

### Step 7: Use the co-function identity
Using the co-function identity \( \sin \theta = \cos(90^\circ - \theta) \), we can equate:
\[ \cos(15^\circ) = \cos(90^\circ - \theta) \]

### Step 8: Solve for \( \theta \)
This implies:
\[ 90^\circ - \theta = 15^\circ \]
Rearranging gives:
\[ \theta = 90^\circ - 15^\circ = 75^\circ \]

### Final Answer
Thus, the value of \( \theta \) is:
\[ \theta = 75^\circ \]