Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation. \( \theta=\frac{\pi}{6} \)

To transform the polar equation \( \theta = \frac{\pi}{6} \) into rectangular coordinates, we can use the relationship between polar and rectangular coordinates. In polar coordinates, \( \theta \) is the angle measured from the positive x-axis, and in rectangular coordinates, we have: - \( x = r \cos(\theta) \) - \( y = r \sin(\theta) \) The equation \( \theta = \frac{\pi}{6} \) represents all points that make an angle of \( \frac{\pi}{6} \) radians (or 30 degrees) with the positive x-axis. This means that the line extends infinitely in both directions along this angle. To express this in rectangular coordinates, we can use the tangent function, which relates the angle to the coordinates: \[ \tan(\theta) = \frac{y}{x} \] Substituting \( \theta = \frac{\pi}{6} \): \[ \tan\left(\frac{\pi}{6}\right) = \frac{y}{x} \] We know that: \[ \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \] Thus, we have: \[ \frac{y}{x} = \frac{1}{\sqrt{3}} \] Multiplying both sides by \( x \) (assuming \( x \neq 0 \)) gives: \[ y = \frac{1}{\sqrt{3}} x \] This is the equation of a line in rectangular coordinates. ### Graphing the Equation The equation \( y = \frac{1}{\sqrt{3}} x \) is a straight line that passes through the origin (0, 0) and has a slope of \( \frac{1}{\sqrt{3}} \). To graph this line: 1. Start at the origin (0, 0). 2. Use the slope \( \frac{1}{\sqrt{3}} \) to find another point. For example, if \( x = \sqrt{3} \), then: \[ y = \frac{1}{\sqrt{3}} \cdot \sqrt{3} = 1 \] So, another point on the line is \( (\sqrt{3}, 1) \). 3. Draw a line through these points, extending infinitely in both directions. The line will make an angle of \( \frac{\pi}{6} \) radians (or 30 degrees) with the positive x-axis, confirming that it represents the same relationship as the original polar equation.