If `pisintheta=1,picostheta=1`, then the value of `{sqrt(3)tan((2)/(3)theta)+1}`is

Options

A. 1
B. `sqrt(3)`
C. 2
D. `(1)/(sqrt(3))`

Answer

To solve the problem, we start with the given equations:

1. \( \pi \sin \theta = 1 \)
2. \( \pi \cos \theta = 1 \)

### Step 1: Solve for \( \sin \theta \) and \( \cos \theta \)

From the first equation, we can express \( \sin \theta \):
\[ \sin \theta = \frac{1}{\pi} \]

From the second equation, we can express \( \cos \theta \):
\[ \cos \theta = \frac{1}{\pi} \]

### Step 2: Find \( \tan \theta \)

Using the definitions of sine and cosine, we can find \( \tan \theta \):
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{1}{\pi}}{\frac{1}{\pi}} = 1 \]

### Step 3: Determine the value of \( \theta \)

Since \( \tan \theta = 1 \), we know that:
\[ \theta = 45^\circ \quad \text{(or } \frac{\pi}{4} \text{ radians)} \]

### Step 4: Calculate \( \frac{2}{3} \theta \)

Now, we calculate \( \frac{2}{3} \theta \):
\[ \frac{2}{3} \theta = \frac{2}{3} \times 45^\circ = 30^\circ \]

### Step 5: Find \( \tan \left( \frac{2}{3} \theta \right) \)

Now we find \( \tan \left( \frac{2}{3} \theta \right) \):
\[ \tan \left( 30^\circ \right) = \frac{1}{\sqrt{3}} \]

### Step 6: Substitute into the expression

Now we substitute \( \tan \left( \frac{2}{3} \theta \right) \) into the expression \( \sqrt{3} \tan \left( \frac{2}{3} \theta \right) + 1 \):
\[ \sqrt{3} \tan \left( \frac{2}{3} \theta \right) + 1 = \sqrt{3} \cdot \frac{1}{\sqrt{3}} + 1 \]

### Step 7: Simplify the expression

This simplifies to:
\[ 1 + 1 = 2 \]

### Final Answer

Thus, the value of \( \sqrt{3} \tan \left( \frac{2}{3} \theta \right) + 1 \) is:
\[ \boxed{2} \]

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