The value of `(sec30^@)/(cosec60^@)` is:

Options

A. `2/3`
B. `sqrt3/2`
C. 1
D. 0

Answer

To solve the expression \(\frac{\sec 30^\circ}{\csc 60^\circ}\), we will first find the values of \(\sec 30^\circ\) and \(\csc 60^\circ\) separately.

### Step 1: Find \(\sec 30^\circ\)

The secant function is defined as the reciprocal of the cosine function:

\[ \sec \theta = \frac{1}{\cos \theta} \]

Thus,

\[ \sec 30^\circ = \frac{1}{\cos 30^\circ} \]

Now, we know that

\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]

So, substituting this value, we get:

\[ \sec 30^\circ = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]

### Step 2: Find \(\csc 60^\circ\)

The cosecant function is defined as the reciprocal of the sine function:

\[ \csc \theta = \frac{1}{\sin \theta} \]

Thus,

\[ \csc 60^\circ = \frac{1}{\sin 60^\circ} \]

Now, we know that

\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]

So, substituting this value, we get:

\[ \csc 60^\circ = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]

### Step 3: Substitute values into the expression

Now that we have both values, we can substitute them into the original expression:

\[ \frac{\sec 30^\circ}{\csc 60^\circ} = \frac{\frac{2}{\sqrt{3}}}{\frac{2}{\sqrt{3}}} \]

### Step 4: Simplify the expression

When we divide the same values, we get:

\[ \frac{\frac{2}{\sqrt{3}}}{\frac{2}{\sqrt{3}}} = 1 \]

### Final Answer

Thus, the value of \(\frac{\sec 30^\circ}{\csc 60^\circ}\) is \(1\).