To convert the given polar coordinates (angle and distance) to rectangular components, we'll use the following formulas:

• $x = r \cdot \cos(\theta)$
• $y = r \cdot \sin(\theta)$

Where:

• $r$ is the distance (16.82 units)
• $\theta$ is the angle (312° 43' 15")

We first need to convert the angle from degrees, minutes, and seconds into decimal degrees: $\theta_{\text{decimal}} = 312 + \frac{43}{60} + \frac{15}{3600}$

Then we will substitute the decimal value of $\theta$ and the distance $r$ into the rectangular coordinate formulas.

Let me compute the values.The angle $\theta$ in decimal degrees is approximately 312.72°.

Using the polar coordinates (distance = 16.82, angle = 312° 43' 15"), the rectangular components are:

• $x \approx 11.41$
• $y \approx -12.36$

Let me know if you'd like more details or have any questions! Here are some related questions:

1. How would the results change if the angle were in radians instead of degrees?
2. What is the general procedure for converting rectangular coordinates back to polar form?
3. How does the cosine and sine function relate to the directional components in navigation (e.g., North-East)?
4. Can the distance ever be negative in polar coordinates, and how would that affect the result?
5. What would the rectangular components be if the angle were 45° instead of 312°?

Tip: Always remember to convert degrees to radians when using trigonometric functions in most programming languages!