If `sintheta sec 23^(@)=1` , the value of `theta` is

Options

A. `23^(@)`
B. `37^(@)`
C. `63^(@)`
D. `67^(@)`

Answer

To solve the equation \( \sin \theta \cdot \sec 23^\circ = 1 \), we can follow these steps:

### Step 1: Rewrite the secant function
The secant function can be rewritten in terms of cosine:
\[ \sec 23^\circ = \frac{1}{\cos 23^\circ} \]
Thus, we can rewrite the equation as:
\[ \sin \theta \cdot \frac{1}{\cos 23^\circ} = 1 \]

### Step 2: Simplify the equation
Multiplying both sides by \( \cos 23^\circ \) gives:
\[ \sin \theta = \cos 23^\circ \]

### Step 3: Use the co-function identity
Using the identity \( \sin(90^\circ - x) = \cos x \), we can express \( \cos 23^\circ \) as:
\[ \cos 23^\circ = \sin(90^\circ - 23^\circ) \]
This simplifies to:
\[ \cos 23^\circ = \sin 67^\circ \]

### Step 4: Set the angles equal
Now we have:
\[ \sin \theta = \sin 67^\circ \]
This implies that:
\[ \theta = 67^\circ \]
since the sine function is positive in the first quadrant.

### Final Answer
Thus, the value of \( \theta \) is:
\[ \theta = 67^\circ \]

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