Find the value of `5sin15^@ sec75^@ + 2tan45^@ + 3sec^(2)30^@`.

Options

A. A)9
B. B)10
C. C)11
D. D)12

Answer

To solve the expression \( 5 \sin 15^\circ \sec 75^\circ + 2 \tan 45^\circ + 3 \sec^2 30^\circ \), we will evaluate each trigonometric function step by step.

### Step 1: Evaluate \( \sec 75^\circ \)

Recall that \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, we can express \( \sec 75^\circ \) as:
\[ \sec 75^\circ = \frac{1}{\cos 75^\circ} \]
Using the identity \( \cos(90^\circ - \theta) = \sin \theta \), we have:
\[ \cos 75^\circ = \sin 15^\circ \]
Thus,
\[ \sec 75^\circ = \frac{1}{\sin 15^\circ} \]

### Step 2: Substitute \( \sec 75^\circ \) into the expression

Now we substitute \( \sec 75^\circ \) back into the expression:
\[ 5 \sin 15^\circ \sec 75^\circ = 5 \sin 15^\circ \cdot \frac{1}{\sin 15^\circ} = 5 \]

### Step 3: Evaluate \( \tan 45^\circ \)

We know that:
\[ \tan 45^\circ = 1 \]
Thus,
\[ 2 \tan 45^\circ = 2 \cdot 1 = 2 \]

### Step 4: Evaluate \( \sec^2 30^\circ \)

Recall that:
\[ \sec \theta = \frac{1}{\cos \theta} \]
So,
\[ \sec 30^\circ = \frac{1}{\cos 30^\circ} \]
And since \( \cos 30^\circ = \frac{\sqrt{3}}{2} \), we have:
\[ \sec 30^\circ = \frac{2}{\sqrt{3}} \]
Now, squaring it gives:
\[ \sec^2 30^\circ = \left(\frac{2}{\sqrt{3}}\right)^2 = \frac{4}{3} \]
Then,
\[ 3 \sec^2 30^\circ = 3 \cdot \frac{4}{3} = 4 \]

### Step 5: Combine all parts

Now, we can combine all the evaluated parts:
\[ 5 + 2 + 4 = 11 \]

### Final Answer

Thus, the value of the expression is:
\[ \boxed{11} \]