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The problem provided is as follows:

You are given the equation \( 3x + 7y = 0 \) with the restriction \( x \leq 0 \). You are asked to:

1. Sketch the least positive angle \( \theta \) for the terminal side of the angle in standard position.
2. Find the values of the six trigonometric functions of \( \theta \).

### Step-by-step solution:

#### 1. Rearrange the equation:
The given equation is \( 3x + 7y = 0 \).

Rearrange to solve for \( y \):
\[
y = -\frac{3}{7}x
\]

This represents a straight line passing through the origin (0, 0) with slope \( -\frac{3}{7} \).

#### 2. Sketching the least positive angle:
Since \( x \leq 0 \), we are concerned with the portion of the line in the second quadrant. The line has a negative slope, meaning it will go downward as you move leftward (since \( x \leq 0 \)).

The terminal side of the angle will be along this line, and we want the least positive angle \( \theta \). The angle \( \theta \) is measured counterclockwise from the positive x-axis to the terminal side.

#### 3. Finding \( \theta \):
The slope of the line is \( -\frac{3}{7} \), so the tangent of the angle \( \theta \) is:
\[
\tan(\theta) = \frac{3}{7}
\]

To find \( \theta \), take the inverse tangent:
\[
\theta = \tan^{-1}\left(\frac{3}{7}\right)
\]

Using a calculator:
\[
\theta \approx 23.2^\circ
\]

However, since the line is in the second quadrant, we must subtract this from \( 180^\circ \):
\[
\theta = 180^\circ - 23.2^\circ = 156.8^\circ
\]

Thus, the least positive angle is approximately \( 156.8^\circ \).

#### 4. Finding the six trigonometric functions:
For the six trigonometric functions, we first note that:

- \( \sin(\theta) = \frac{3}{\sqrt{3^2 + 7^2}} = \frac{3}{\sqrt{58}} \)
- \( \cos(\theta) = \frac{-7}{\sqrt{3^2 + 7^2}} = \frac{-7}{\sqrt{58}} \)
- \( \tan(\theta) = \frac{3}{7} \)
- \( \csc(\theta) = \frac{\sqrt{58}}{3} \)
- \( \sec(\theta) = \frac{-\sqrt{58}}{7} \)
- \( \cot(\theta) = \frac{7}{3} \)

These are the six trigonometric function values for the angle \( \theta \approx 156.8^\circ \).

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Do you have any questions about this, or would you like more details?

### Related Questions:
1. How is the reference angle calculated for an angle in the second quadrant?
2. How does the slope of the line relate to the tangent of the angle?
3. Why do we subtract from \( 180^\circ \) to get the angle in the second quadrant?
4. What are the characteristics of angles in the second quadrant?
5. How can we use the unit circle to find the sine and cosine of angles in different quadrants?

### Tip:
Always remember the signs of trigonometric functions in different quadrants: sine is positive in the second quadrant, while cosine and tangent are negative.

An equation shown below for the terminal side of an angle θ in standard position is given with a restriction on x: 3x + 7y = 0, x ≤ 0. Sketch the least positive angle θ, and find the values of the six trigonometric functions of θ.

Learn how to find the least positive angle θ for the line 3x + 7y = 0, restricted to x ≤ 0, and calculate the six trigonometric functions. This high school trigonometry problem involves inverse tangent and quadrant rules, perfect for students in Grades 10-12.

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