Inates of the point P on the rcumference of the circle. (Hint: Add x- nd y-axes, assuming that the angle is in Candard position.) The coordinates of P are P 10 120°
Inates of the point P on the rcumference of the circle. (Hint: Add x- nd y-axes, assuming that the angle is in Candard position.) The coordinates of P are P 10 120°
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
![**Problem Statement:**
Find the coordinates of the point P on the circumference of the circle. (Hint: Add x- and y-axes, assuming that the angle is in standard position.)
**Diagram Description:**
The diagram depicts a circle with a radius of 10 units. There is a central angle of 120° subtended by the radius lines at the center of the circle. Point P is located on the circumference of the circle such that the angle between the positive x-axis and the line segment connecting the center of the circle to point P is 120°.
**Solution:**
To find the coordinates of point P, we utilize the information provided:
1. The radius \( r \) of the circle is 10 units.
2. The angle \( \theta \), measured from the positive x-axis, is 120°.
Using the polar-to-Cartesian coordinate transformation formulas:
- \( x = r \cos(\theta) \)
- \( y = r \sin(\theta) \)
We substitute \( r = 10 \) and \( \theta = 120° \) into these formulas.
We know from trigonometry:
- \( \cos(120°) = -\frac{1}{2} \)
- \( \sin(120°) = \frac{\sqrt{3}}{2} \)
Plugging these values in:
\[ x = 10 \cos(120°) = 10 \left( -\frac{1}{2} \right) = -5 \]
\[ y = 10 \sin(120°) = 10 \left( \frac{\sqrt{3}}{2} \right) = 5\sqrt{3} \]
Therefore, the coordinates of point P are:
\[ \boxed{(-5, 5\sqrt{3})} \]
**Conclusion:**
Using trigonometric identities and transformation from polar to Cartesian coordinates, we have determined the coordinates of point P on the circumference of the circle. The answer is expressed in its simplest form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c3214b5-f2f5-4a89-904a-2906edf42c76%2Fad043168-f38f-4497-9477-05330ee747e6%2Fs2mhy4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the coordinates of the point P on the circumference of the circle. (Hint: Add x- and y-axes, assuming that the angle is in standard position.)
**Diagram Description:**
The diagram depicts a circle with a radius of 10 units. There is a central angle of 120° subtended by the radius lines at the center of the circle. Point P is located on the circumference of the circle such that the angle between the positive x-axis and the line segment connecting the center of the circle to point P is 120°.
**Solution:**
To find the coordinates of point P, we utilize the information provided:
1. The radius \( r \) of the circle is 10 units.
2. The angle \( \theta \), measured from the positive x-axis, is 120°.
Using the polar-to-Cartesian coordinate transformation formulas:
- \( x = r \cos(\theta) \)
- \( y = r \sin(\theta) \)
We substitute \( r = 10 \) and \( \theta = 120° \) into these formulas.
We know from trigonometry:
- \( \cos(120°) = -\frac{1}{2} \)
- \( \sin(120°) = \frac{\sqrt{3}}{2} \)
Plugging these values in:
\[ x = 10 \cos(120°) = 10 \left( -\frac{1}{2} \right) = -5 \]
\[ y = 10 \sin(120°) = 10 \left( \frac{\sqrt{3}}{2} \right) = 5\sqrt{3} \]
Therefore, the coordinates of point P are:
\[ \boxed{(-5, 5\sqrt{3})} \]
**Conclusion:**
Using trigonometric identities and transformation from polar to Cartesian coordinates, we have determined the coordinates of point P on the circumference of the circle. The answer is expressed in its simplest form.
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