Find the equation to the straight line, whose perpendicular distance from the origin is 4 and perpendicular makes angle - π/6 positive x-axis.

To find the equation of the straight line whose perpendicular distance from the origin is 4 units, and where the perpendicular makes an angle of \(-\frac{\pi}{6}\) with the positive x-axis, we can use the normal form of the equation of a straight line.

### Normal Form of the Equation of a Straight Line:
The general equation of a line in the normal form is given by:

\[
x \cos \theta + y \sin \theta = p
\]

Where:
- \( p \) is the perpendicular distance from the origin to the line.
- \( \theta \) is the angle that the perpendicular makes with the positive x-axis.

### Given Values:
- \( p = 4 \)
- \( \theta = -\frac{\pi}{6} \)

### Substituting into the Equation:
Now, we substitute these values into the normal form equation:

\[
x \cos\left(-\frac{\pi}{6}\right) + y \sin\left(-\frac{\pi}{6}\right) = 4
\]

### Simplifying the Trigonometric Values:
- \( \cos\left(-\frac{\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \)
- \( \sin\left(-\frac{\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) = -\frac{1}{2} \)

Substituting these values:

\[
x \cdot \frac{\sqrt{3}}{2} + y \cdot \left(-\frac{1}{2}\right) = 4
\]

### Final Equation:
Multiply through by 2 to eliminate the fractions:

\[
\sqrt{3}x - y = 8
\]

So, the equation of the straight line is:

\[
\sqrt{3}x - y = 8
\]

Would you like more details or have any questions?

### Follow-up Questions:
1. What is the significance of the normal form of a line equation?
2. How would the equation change if the angle were \(\frac{\pi}{6}\) instead of \(-\frac{\pi}{6}\)?
3. What if the perpendicular distance from the origin was different, say 5 units?
4. How does the angle \( \theta \) affect the orientation of the line?
5. Can you derive the line equation using another form, such as point-slope form?
6. How would the equation change if the line were parallel to the y-axis?
7. What happens to the equation if the angle is 0?
8. How does the slope-intercept form of the equation relate to this one?

### Tip:
Understanding how different forms of line equations relate to each other can help you solve problems more flexibly.

Discover how to find the equation of a straight line whose perpendicular distance from the origin is 4 units. Learn the application of trigonometry and the normal form of a line equation. Suitable for high school students studying coordinate geometry and trigonometry.

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