To find the slope-intercept form of the equation of the line that passes through the point $P(-9, -8)$ and makes an angle $\theta = 150^\circ$ with the positive x-axis, follow these steps:

Step 1: Determine the slope

The slope $m$ of a line that makes an angle $\theta$ with the positive x-axis can be found using the tangent function: $m = \tan(\theta)$

For $\theta = 150^\circ$: $m = \tan(150^\circ)$

Using the trigonometric identity: $\tan(150^\circ) = \tan(180^\circ - 30^\circ) = -\tan(30^\circ)$ $\tan(30^\circ) = \frac{1}{\sqrt{3}}$

So, $\tan(150^\circ) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$

Thus, the slope $m$ is: $m = -\frac{\sqrt{3}}{3}$

Step 2: Use the point-slope form to find the equation of the line

The point-slope form of the equation of a line is: $y - y_1 = m(x - x_1)$

Where $(x_1, y_1)$ is the point the line passes through, and $m$ is the slope. For the point $P(-9, -8)$: $y - (-8) = -\frac{\sqrt{3}}{3}(x - (-9))$

Simplify the equation: $y + 8 = -\frac{\sqrt{3}}{3}(x + 9)$

Step 3: Convert to slope-intercept form $y = mx + b$

Distribute the slope on the right side: $y + 8 = -\frac{\sqrt{3}}{3}x - \frac{\sqrt{3}}{3} \cdot 9$

Simplify further: $y + 8 = -\frac{\sqrt{3}}{3}x - 3\sqrt{3}$

Subtract 8 from both sides to solve for $y$: $y = -\frac{\sqrt{3}}{3}x - 3\sqrt{3} - 8$

Thus, the slope-intercept form of the equation of the line is: $y = -\frac{\sqrt{3}}{3}x - 3\sqrt{3} - 8$

Would you like further details or have any questions about this solution?

Relative Questions:

1. How can we derive the tangent of an angle using trigonometric identities?
2. What are the properties of lines with angles greater than $90^\circ$ with respect to the x-axis?
3. How does changing the point through which the line passes affect the slope-intercept form?
4. Can we use a different method to find the equation of the line, such as using vectors?
5. What are the implications of having a negative slope for the line's graph?

Tip:

When working with angles and slopes, remember that the tangent function is periodic with a period of $180^\circ$, which helps in determining the slope for various angles.