Question

two points are given in polar coordinates by (r, θ) = (1.60 m, 50.0°) and (r, θ) = (4.00 m, −38.0°), respectively. what is the distance between them? m

Explanation:

Step1: Recall the polar distance formula

The distance \( d \) between two points in polar coordinates \((r_1, \theta_1)\) and \((r_2, \theta_2)\) is given by the law of cosines: \( d = \sqrt{r_1^2 + r_2^2 - 2r_1r_2\cos(\theta_1 - \theta_2)} \).

Step2: Identify the values

Here, \( r_1 = 1.60 \, \text{m} \), \( \theta_1 = 50.0^\circ \), \( r_2 = 4.00 \, \text{m} \), \( \theta_2 = -38.0^\circ \). First, calculate the angle difference: \( \Delta\theta = \theta_1 - \theta_2 = 50.0^\circ - (-38.0^\circ) = 88.0^\circ \).

Step3: Substitute into the formula

Substitute \( r_1 = 1.60 \), \( r_2 = 4.00 \), and \( \Delta\theta = 88.0^\circ \) (convert to radians if needed, but most calculators can handle degrees for cosine) into the formula:
\[
d = \sqrt{(1.60)^2 + (4.00)^2 - 2(1.60)(4.00)\cos(88.0^\circ)}
\]
Calculate each term:

• \( (1.60)^2 = 2.56 \)
• \( (4.00)^2 = 16.00 \)
• \( 2(1.60)(4.00) = 12.80 \)
• \( \cos(88.0^\circ) \approx 0.0349 \) (using calculator)

Now, compute the term inside the square root:
\[
2.56 + 16.00 - 12.80 \times 0.0349 \approx 2.56 + 16.00 - 0.4467 \approx 18.1133
\]

Step4: Take the square root

\[
d = \sqrt{18.1133} \approx 4.256 \, \text{m}
\]

Answer:

\( \approx 4.26 \, \text{m} \) (rounded to three significant figures)