What is the value of `(3)/(2)((cos39)/(sin51))-sqrt(sin^(2)39+sin^(2)51)` ?

Options

A. `(1)/(2)`
B. `(5)/(2)`
C. 0
D. Both `(1)/(2)and(5)/(2)`

Answer

To solve the question, we need to evaluate the expression:

\[ \frac{3}{2} \left( \frac{\cos 39^\circ}{\sin 51^\circ} \right) - \sqrt{\sin^2 39^\circ + \sin^2 51^\circ} \]

### Step 1: Simplify \(\frac{\cos 39^\circ}{\sin 51^\circ}\)

Using the complementary angle identity, we know that:

\[ \sin 51^\circ = \cos 39^\circ \]

Thus, we can rewrite the fraction:

\[ \frac{\cos 39^\circ}{\sin 51^\circ} = \frac{\cos 39^\circ}{\cos 39^\circ} = 1 \]

### Step 2: Substitute back into the expression

Now substituting this back into the original expression:

\[ \frac{3}{2} \cdot 1 - \sqrt{\sin^2 39^\circ + \sin^2 51^\circ} \]

This simplifies to:

\[ \frac{3}{2} - \sqrt{\sin^2 39^\circ + \sin^2 51^\circ} \]

### Step 3: Evaluate \(\sin^2 39^\circ + \sin^2 51^\circ\)

Using the identity for complementary angles again, we have:

\[ \sin^2 51^\circ = \cos^2 39^\circ \]

Thus:

\[ \sin^2 39^\circ + \sin^2 51^\circ = \sin^2 39^\circ + \cos^2 39^\circ = 1 \]

### Step 4: Substitute back into the expression

Now substituting this value back into the expression:

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

This simplifies to:

\[ \frac{3}{2} - 1 \]

### Step 5: Final calculation

Calculating the final value:

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

Thus, the final answer is:

\[ \frac{1}{2} \]