Find the equation of the st.line through the origin making angle of `60^(@)` with st. Line `x+sqrt(3)y+3sqrt(3)=0`.

Answer

To find the equation of the straight line through the origin that makes an angle of \(60^\circ\) with the line given by the equation \(x + \sqrt{3}y + 3\sqrt{3} = 0\), we can follow these steps:

### Step 1: Find the slope of the given line
The given line can be rewritten in slope-intercept form \(y = mx + b\). Starting from the equation:
\[ x + \sqrt{3}y + 3\sqrt{3} = 0 \]
we can isolate \(y\):
\[ \sqrt{3}y = -x - 3\sqrt{3} \]
\[ y = -\frac{1}{\sqrt{3}}x - 3 \]
From this, we can see that the slope \(m_1\) of the given line is:
\[ m_1 = -\frac{1}{\sqrt{3}} \]

### Step 2: Use the angle formula to find the slope of the required line
We know that the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is given by:
\[ \tan(\theta) = \frac{m_1 - m_2}{1 + m_1 m_2} \]
Given that \(\theta = 60^\circ\), we have:
\[ \tan(60^\circ) = \sqrt{3} \]
Substituting \(m_1\) and \(\tan(60^\circ)\) into the formula gives:
\[ \sqrt{3} = \frac{-\frac{1}{\sqrt{3}} - m_2}{1 + \left(-\frac{1}{\sqrt{3}}\right)m_2} \]

### Step 3: Solve for \(m_2\)
Cross-multiplying to eliminate the fraction:
\[ \sqrt{3}(1 - \frac{1}{\sqrt{3}}m_2) = -\frac{1}{\sqrt{3}} - m_2 \]
Expanding this:
\[ \sqrt{3} - m_2 = -\frac{1}{\sqrt{3}} - m_2 \]
Rearranging gives:
\[ \sqrt{3} + \frac{1}{\sqrt{3}} = 0 \]
This leads to:
\[ m_2 = \sqrt{3} + \frac{1}{\sqrt{3}}m_2 \]
Solving for \(m_2\):
\[ m_2(\sqrt{3} + 1) = \sqrt{3} \]
\[ m_2 = \frac{\sqrt{3}}{\sqrt{3} + 1} \]

### Step 4: Write the equation of the line through the origin
Using the point-slope form of the line equation \(y - y_1 = m(x - x_1)\) and knowing the line passes through the origin \((0, 0)\):
\[ y - 0 = m_2(x - 0) \]
Thus, the equation simplifies to:
\[ y = m_2 x \]

### Step 5: Substitute \(m_2\) to get the final equation
Substituting \(m_2\):
\[ y = \frac{\sqrt{3}}{\sqrt{3} + 1} x \]

### Final Answer
The equation of the required straight line is:
\[ y = \frac{\sqrt{3}}{\sqrt{3} + 1} x \]