Let's analyze the problem from the image you shared.

Problem Breakdown:

1. Equation: $-6x - 2y = 0$, with the restriction $x \leq 0$.
2. The goal is to:
   • Find the least positive angle $\theta$ for which this equation represents the terminal side of the angle in standard position.
   • Determine the values of the six trigonometric functions for $\theta$.
   • Choose the graph corresponding to this equation with the least positive angle $\theta$.

Solving for the Equation and the Angle:

1. Rewriting the equation: $-6x - 2y = 0$ can be rewritten as: $2y = -6x$ $y = -3x$ This is a straight line with a slope of $-3$, passing through the origin (since there's no constant term).

2. Identifying the angle:

   • The slope of the line is $-3$, so the terminal side of the angle lies in Quadrant II because the restriction is $x \leq 0$ (i.e., we are looking for a graph that covers negative $x$).
   • The angle $\theta$ can be found using the inverse tangent function. The slope $-3$ corresponds to: $\theta = \tan^{-1}(-3)$ which will give an angle in Quadrant II.

Identifying the Graph:

• Quadrant II behavior means we expect the line to rise as it moves to the left along the $x$-axis (i.e., $x \leq 0$).
• Let's match the graphs with this understanding:
  • Option A: Shows a line in Quadrant IV, which does not match our expectations.
  • Option B: Shows a line in Quadrant II, which fits our equation's behavior.
  • Option C: Shows a parabola, which does not match our equation (since it's linear).

Thus, Option B is the correct graph for the given equation.

Would you like further details on how to calculate the six trigonometric functions or any additional clarifications?

Related Questions:

1. How do you calculate the six trigonometric functions given an angle?
2. What is the process for finding the angle using the inverse tangent?
3. How does the slope of a line relate to its angle in standard position?
4. What are the typical signs of trigonometric functions in different quadrants?
5. How would the graph change if the restriction on $x$ were different?

Tip:

When finding the angle for a line, always check its slope and consider which quadrant the line passes through based on restrictions like $x \leq 0$ or $x \geq 0$.