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.) 10 The coordinates of P are (Type an ordered pair. Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) 120°
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.) 10 The coordinates of P are (Type an ordered pair. Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) 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°.
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![### Finding the Coordinates of a Point on a Circle
To find the coordinates of the point P on the circumference of the circle, follow these steps (Hint: Add x- and y-axes, assuming that the angle is in standard position).
1. **Understanding the Problem:**
- **Circle Information:** The radius of the circle is 10 units.
- **Angle Information:** The angle is given as 120 degrees.
2. **Graph Description:**
- **Diagram:** A circle is drawn with a radius extending from the center to point P.
- **Radius (r):** The length of the radius is labeled as 10 units.
- **Angle (θ):** The angle between the radius and the x-axis, measured counterclockwise, is labeled as 120 degrees.
3. **Calculating Coordinates:**
- **Using Trigonometric Functions:**
- The x-coordinate of P can be found using \( x = r \cos(\theta) \).
- The y-coordinate of P can be found using \( y = r \sin(\theta) \).
- **Substitute Values:**
- \( r = 10 \)
- \( \theta = 120^\circ \)
- \( x = 10 \cos(120^\circ) \)
- \( y = 10 \sin(120^\circ) \)
- **Simplify Using Trigonometric Values:**
- \(\cos(120^\circ) = -\frac{1}{2} \)
- \(\sin(120^\circ) = \frac{\sqrt{3}}{2} \)
- **Final Coordinates:**
- \( x = 10 \left(-\frac{1}{2}\right) = -5 \)
- \( y = 10 \left(\frac{\sqrt{3}}{2}\right) = 5\sqrt{3} \)
Therefore, the coordinates of point P are \(\left( -5, 5\sqrt{3} \right)\).
**Boxed Answer:**
\[
\text{The coordinates of P are } \left( -5, 5\sqrt{3} \right).
\]
*(Type an ordered pair. Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb910418a-e376-48de-ad2c-cad9a3f8e771%2F943fe9b0-fadb-4540-ac3f-d77e6f25fc5b%2F1tw2lsbi_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the Coordinates of a Point on a Circle
To find the coordinates of the point P on the circumference of the circle, follow these steps (Hint: Add x- and y-axes, assuming that the angle is in standard position).
1. **Understanding the Problem:**
- **Circle Information:** The radius of the circle is 10 units.
- **Angle Information:** The angle is given as 120 degrees.
2. **Graph Description:**
- **Diagram:** A circle is drawn with a radius extending from the center to point P.
- **Radius (r):** The length of the radius is labeled as 10 units.
- **Angle (θ):** The angle between the radius and the x-axis, measured counterclockwise, is labeled as 120 degrees.
3. **Calculating Coordinates:**
- **Using Trigonometric Functions:**
- The x-coordinate of P can be found using \( x = r \cos(\theta) \).
- The y-coordinate of P can be found using \( y = r \sin(\theta) \).
- **Substitute Values:**
- \( r = 10 \)
- \( \theta = 120^\circ \)
- \( x = 10 \cos(120^\circ) \)
- \( y = 10 \sin(120^\circ) \)
- **Simplify Using Trigonometric Values:**
- \(\cos(120^\circ) = -\frac{1}{2} \)
- \(\sin(120^\circ) = \frac{\sqrt{3}}{2} \)
- **Final Coordinates:**
- \( x = 10 \left(-\frac{1}{2}\right) = -5 \)
- \( y = 10 \left(\frac{\sqrt{3}}{2}\right) = 5\sqrt{3} \)
Therefore, the coordinates of point P are \(\left( -5, 5\sqrt{3} \right)\).
**Boxed Answer:**
\[
\text{The coordinates of P are } \left( -5, 5\sqrt{3} \right).
\]
*(Type an ordered pair. Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)*
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