Paove that the distane, b, points with between two polar coordinates (s, Bq) aud (s2,E2) is bvs+s-2ss2 cos (02-8, COS
Paove that the distane, b, points with between two polar coordinates (s, Bq) aud (s2,E2) is bvs+s-2ss2 cos (02-8, COS
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
Related questions
Question
![**Title: Calculating the Distance Between Two Points in Polar Coordinates**
**Objective:**
To prove that the distance, \( b \), between two points with polar coordinates \((s_1, \theta_1)\) and \((s_2, \theta_2)\) is given by the formula:
\[
b = \sqrt{s_1^2 + s_2^2 - 2s_1s_2 \cos(\theta_2 - \theta_1)}
\]
**Explanation:**
Polar coordinates represent a point in terms of a radius and an angle from the positive x-axis. Here, \((s_1, \theta_1)\) and \((s_2, \theta_2)\) are two points where:
- \(s_1\) and \(s_2\) are the radii of the points from the origin.
- \(\theta_1\) and \(\theta_2\) are the angles made with the positive x-axis.
**Formula Derivation:**
1. Start with the Cartesian coordinates formula for the distance between two points:
\[
c = \sqrt{{(x_2 - x_1)}^2 + {(y_2 - y_1)}^2}
\]
2. Convert polar to Cartesian coordinates:
- \(x = s \cos(\theta)\)
- \(y = s \sin(\theta)\)
3. Substitute polar coordinates:
- \(x_1 = s_1 \cos(\theta_1)\), \(y_1 = s_1 \sin(\theta_1)\)
- \(x_2 = s_2 \cos(\theta_2)\), \(y_2 = s_2 \sin(\theta_2)\)
4. Compute the distance:
\[
b = \sqrt{{(s_2 \cos(\theta_2) - s_1 \cos(\theta_1))}^2 + {(s_2 \sin(\theta_2) - s_1 \sin(\theta_1))}^2}
\]
5. Simplify to show that it is equivalent to:
\[
b = \sqrt{s_1^2 + s_2^2 - 2s_1s_2 \cos(\theta_2 - \theta_1)}
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Transcribed Image Text:**Title: Calculating the Distance Between Two Points in Polar Coordinates**
**Objective:**
To prove that the distance, \( b \), between two points with polar coordinates \((s_1, \theta_1)\) and \((s_2, \theta_2)\) is given by the formula:
\[
b = \sqrt{s_1^2 + s_2^2 - 2s_1s_2 \cos(\theta_2 - \theta_1)}
\]
**Explanation:**
Polar coordinates represent a point in terms of a radius and an angle from the positive x-axis. Here, \((s_1, \theta_1)\) and \((s_2, \theta_2)\) are two points where:
- \(s_1\) and \(s_2\) are the radii of the points from the origin.
- \(\theta_1\) and \(\theta_2\) are the angles made with the positive x-axis.
**Formula Derivation:**
1. Start with the Cartesian coordinates formula for the distance between two points:
\[
c = \sqrt{{(x_2 - x_1)}^2 + {(y_2 - y_1)}^2}
\]
2. Convert polar to Cartesian coordinates:
- \(x = s \cos(\theta)\)
- \(y = s \sin(\theta)\)
3. Substitute polar coordinates:
- \(x_1 = s_1 \cos(\theta_1)\), \(y_1 = s_1 \sin(\theta_1)\)
- \(x_2 = s_2 \cos(\theta_2)\), \(y_2 = s_2 \sin(\theta_2)\)
4. Compute the distance:
\[
b = \sqrt{{(s_2 \cos(\theta_2) - s_1 \cos(\theta_1))}^2 + {(s_2 \sin(\theta_2) - s_1 \sin(\theta_1))}^2}
\]
5. Simplify to show that it is equivalent to:
\[
b = \sqrt{s_1^2 + s_2^2 - 2s_1s_2 \cos(\theta_2 - \theta_1)}
\]
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