If r `sintheta=1,r costheta=sqrt(3)` then the value of `(r^(2)tantheta)` is

Options

A. 4
B. `(1)/(sqrt(3))`
C. `(4)/(sqrt(3))`
D. `4sqrt(3)

Answer

To solve the problem, we have the following equations:

1. \( r \sin \theta = 1 \)
2. \( r \cos \theta = \sqrt{3} \)

We need to find the value of \( r^2 \tan \theta \).

### Step 1: Square both equations

First, we will square both equations:

- From \( r \sin \theta = 1 \):
\[ (r \sin \theta)^2 = 1^2 \implies r^2 \sin^2 \theta = 1 \]

- From \( r \cos \theta = \sqrt{3} \):
\[ (r \cos \theta)^2 = (\sqrt{3})^2 \implies r^2 \cos^2 \theta = 3 \]

### Step 2: Add the squared equations

Now, we will add both equations:

\[ r^2 \sin^2 \theta + r^2 \cos^2 \theta = 1 + 3 \]

This simplifies to:

\[ r^2 (\sin^2 \theta + \cos^2 \theta) = 4 \]

### Step 3: Use the Pythagorean identity

Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \):

\[ r^2 \cdot 1 = 4 \implies r^2 = 4 \]

### Step 4: Find \( \tan \theta \)

Now, we can find \( \tan \theta \):

\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]

From the first equation \( r \sin \theta = 1 \), we have:

\[ \sin \theta = \frac{1}{r} = \frac{1}{2} \quad \text{(since \( r^2 = 4 \) implies \( r = 2 \))} \]

From the second equation \( r \cos \theta = \sqrt{3} \):

\[ \cos \theta = \frac{\sqrt{3}}{r} = \frac{\sqrt{3}}{2} \]

Thus, we can find \( \tan \theta \):

\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \]

### Step 5: Calculate \( r^2 \tan \theta \)

Now, we can find \( r^2 \tan \theta \):

\[ r^2 \tan \theta = 4 \cdot \frac{1}{\sqrt{3}} = \frac{4}{\sqrt{3}} \]

### Final Answer

Thus, the value of \( r^2 \tan \theta \) is \( \frac{4}{\sqrt{3}} \).

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