What is the value of `(3)/(2)((cos 39)/(sin 51))-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 expression \(\frac{3}{2}\left(\frac{\cos 39^\circ}{\sin 51^\circ}\right) - \sqrt{\sin^2 39^\circ + \sin^2 51^\circ}\), we can follow these steps:

### Step 1: Simplify \(\sin 51^\circ\)
Using the co-function identity, we know that:
\[ \sin 51^\circ = \sin(90^\circ - 39^\circ) = \cos 39^\circ \]
Thus, we can rewrite the expression:
\[ \frac{3}{2}\left(\frac{\cos 39^\circ}{\cos 39^\circ}\right) - \sqrt{\sin^2 39^\circ + \sin^2 51^\circ} \]

### Step 2: Simplify the fraction
Since \(\frac{\cos 39^\circ}{\cos 39^\circ} = 1\), we have:
\[ \frac{3}{2}(1) - \sqrt{\sin^2 39^\circ + \sin^2 51^\circ} \]
This simplifies to:
\[ \frac{3}{2} - \sqrt{\sin^2 39^\circ + \sin^2 51^\circ} \]

### Step 3: Simplify \(\sin^2 51^\circ\)
Now, substituting \(\sin 51^\circ\) with \(\cos 39^\circ\):
\[ \sin^2 51^\circ = \cos^2 39^\circ \]
Thus, we can write:
\[ \sin^2 39^\circ + \sin^2 51^\circ = \sin^2 39^\circ + \cos^2 39^\circ \]

### Step 4: Apply the Pythagorean identity
Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\):
\[ \sin^2 39^\circ + \cos^2 39^\circ = 1 \]
Now we substitute this back into the expression:
\[ \frac{3}{2} - \sqrt{1} \]

### Step 5: Calculate the square root
Since \(\sqrt{1} = 1\), we have:
\[ \frac{3}{2} - 1 \]

### Step 6: Simplify the final expression
Now, simplifying \(\frac{3}{2} - 1\):
\[ \frac{3}{2} - \frac{2}{2} = \frac{1}{2} \]

### Final Result
Thus, the value of the expression is:
\[ \frac{1}{2} \]