Let's start by visualizing the problem with a diagram. We have a point [tex]\(P(5, -2)\)[/tex] lying in a Cartesian coordinate system. To find the angle [tex]\(\theta\)[/tex] in standard position, we will use geometric reasoning and trigonometric principles.
### Diagram
1. Draw the coordinate system:
- Label the x-axis horizontally and the y-axis vertically.
- Mark the origin [tex]\(O (0, 0)\)[/tex].
2. Plot the point [tex]\(P(5, -2)\)[/tex]:
- Move 5 units to the right from the origin (positive x-direction).
- Move 2 units down from there (negative y-direction).
- Mark the point [tex]\(P\)[/tex].
3. Draw the terminal arm:
- Draw a straight line from the origin [tex]\(O\)[/tex] to the point [tex]\(P\)[/tex].
4. Indicate the angle [tex]\(\theta\)[/tex]:
- The angle [tex]\(\theta\)[/tex] is measured counterclockwise from the positive x-axis to the terminal arm.
### Finding the Related Acute Angle
The related acute angle is the angle formed by the terminal arm with the x-axis within the triangle formed, ignoring in which quadrant the terminal arm lies. To find this angle, we use the tangent function because we know the opposite side ([tex]\(y = -2\)[/tex]) and the adjacent side ([tex]\(x = 5\)[/tex]).
[tex]\[
\text{tan}(\text{related angle}) = \left| \frac{y}{x} \right| = \left| \frac{-2}{5} \right| = \frac{2}{5}
\][/tex]
The related acute angle ([tex]\(\alpha\)[/tex]) can be found by taking the arctangent (inverse tangent):
[tex]\[
\alpha = \arctan\left(\frac{2}{5}\right)
\][/tex]
This gives us the related acute angle to be approximately [tex]\(21.801\)[/tex] degrees.
Rounding to the nearest degree:
[tex]\[
\alpha \approx 22^{\circ}
\][/tex]
### Determining the Terminal Angle
To determine the terminal angle [tex]\(\theta\)[/tex], we need to consider in which quadrant the point [tex]\(P\)[/tex] lies. The coordinates [tex]\( (5, -2) \)[/tex] place point [tex]\( P \)[/tex] in the fourth quadrant.
In the fourth quadrant, the terminal angle ([tex]\(\theta\)[/tex]) can be calculated using:
[tex]\[
\theta = 360^{\circ} - \text{related angle}
\][/tex]
Thus:
[tex]\[
\theta = 360^{\circ} - 21.801^{\circ} \approx 338^{\circ}
\][/tex]
Rounding to the nearest degree:
[tex]\[
\theta \approx 338^{\circ}
\][/tex]
### Summary
- a. The related acute angle is [tex]\(\mathbf{22^{\circ}}\)[/tex].
- b. The terminal angle is [tex]\(\mathbf{338^{\circ}}\)[/tex].
These steps and outcomes correctly determine the related acute angle and the terminal angle for the point [tex]\(P(5, -2)\)[/tex].
