if `(1-costheta)/(sintheta)=(sqrt(3))/(3)`, then `theta`=

Options

A. `15^(@)`
B. `30^(@)`
C. `45^(@)`
D. `60^(@)`

Explanation

To solve the equation \(\frac{1 - \cos \theta}{\sin \theta} = \frac{\sqrt{3}}{3}\), we will follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ \frac{1 - \cos \theta}{\sin \theta} = \frac{\sqrt{3}}{3} \]

### Step 2: Cross-multiply
Cross-multiply to eliminate the fraction:
\[ 1 - \cos \theta = \frac{\sqrt{3}}{3} \sin \theta \]

### Step 3: Rearrange the equation
Rearranging gives:
\[ 1 - \cos \theta = \frac{\sqrt{3}}{3} \sin \theta \]

### Step 4: Isolate the trigonometric terms
We can express this as:
\[ \frac{1}{\sin \theta} - \frac{\cos \theta}{\sin \theta} = \frac{1}{\sqrt{3}} \]
This can be rewritten using cosecant and cotangent:
\[ \csc \theta - \cot \theta = \frac{1}{\sqrt{3}} \]

### Step 5: Use trigonometric identities
Using the identity \(\csc \theta - \cot \theta = \tan \frac{\theta}{2}\), we can rewrite the equation:
\[ \tan \frac{\theta}{2} = \frac{1}{\sqrt{3}} \]

### Step 6: Solve for \(\frac{\theta}{2}\)
The value of \(\tan \frac{\theta}{2} = \frac{1}{\sqrt{3}}\) corresponds to an angle of \(30^\circ\):
\[ \frac{\theta}{2} = 30^\circ \]

### Step 7: Solve for \(\theta\)
Now, multiply both sides by 2 to find \(\theta\):
\[ \theta = 60^\circ \]

### Final Answer
Thus, the value of \(\theta\) is:
\[ \theta = 60^\circ \]

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