A = 5π/3 Find the exact value (no rounding) of sine. 5п sin 3 O Type here to search At a
A = 5π/3 Find the exact value (no rounding) of sine. 5п sin 3 O Type here to search At a
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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Question
![**Transcription:**
The graph below shows the angle \( A = \frac{5\pi}{3} \) inside the unit circle.
[Image Description: The diagram presents a unit circle (a circle with radius 1) centered at the origin of the coordinate plane. An angle of \( A = \frac{5\pi}{3} \) radians is drawn inside the circle. One arm of the angle lies along the positive x-axis, with the other arm extending into the fourth quadrant. A right triangle is formed by dropping a perpendicular from the point on the circle to the x-axis. The hypotenuse of the right triangle is the radius of the unit circle. The sides of the triangle include the red line showing the vertical component and the blue line showing the horizontal component.]
Find the exact value (no rounding) of sine.
\[ \sin\left(\frac{5\pi}{3}\right) = \]
[Text box for input]
**Explanation of Diagram:**
- **Unit Circle**: A circle with a radius of 1, centered at the origin of the coordinate plane.
- **Angle \( A = \frac{5\pi}{3} \)**: The angle is measured from the positive x-axis in the counter-clockwise direction. This angle is equivalent to -60 degrees (or \( 300^\circ \)) when measured clockwise.
- **Right Triangle**: The right triangle formed includes:
- **Hypotenuse**: The radius of the unit circle, which is equal to 1.
- **Vertical Component (Red Line)**: Represents the sine of the angle.
- **Horizontal Component (Blue Line)**: Represents the cosine of the angle.
The task is to determine the exact sine value of the angle \( \frac{5\pi}{3} \) without rounding.
**Solution Steps:**
1. Recall that the unit circle coordinates for an angle \( \theta \) are given by \( (\cos(\theta), \sin(\theta)) \).
2. For angle \( \frac{5\pi}{3} \) radians, which is equivalent to \( -\frac{\pi}{3} \) radians:
- The reference angle in the fourth quadrant is \( \frac{\pi}{3} \).
- The cosine of \( \frac{5\pi}{3} \) is positive, while the sine is negative.
Therefore, \( \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc1123ec-a0e0-4e35-afe0-df78f47a58b5%2F52abfb10-4d4e-4086-89d4-706725c6b1f7%2Frags2p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription:**
The graph below shows the angle \( A = \frac{5\pi}{3} \) inside the unit circle.
[Image Description: The diagram presents a unit circle (a circle with radius 1) centered at the origin of the coordinate plane. An angle of \( A = \frac{5\pi}{3} \) radians is drawn inside the circle. One arm of the angle lies along the positive x-axis, with the other arm extending into the fourth quadrant. A right triangle is formed by dropping a perpendicular from the point on the circle to the x-axis. The hypotenuse of the right triangle is the radius of the unit circle. The sides of the triangle include the red line showing the vertical component and the blue line showing the horizontal component.]
Find the exact value (no rounding) of sine.
\[ \sin\left(\frac{5\pi}{3}\right) = \]
[Text box for input]
**Explanation of Diagram:**
- **Unit Circle**: A circle with a radius of 1, centered at the origin of the coordinate plane.
- **Angle \( A = \frac{5\pi}{3} \)**: The angle is measured from the positive x-axis in the counter-clockwise direction. This angle is equivalent to -60 degrees (or \( 300^\circ \)) when measured clockwise.
- **Right Triangle**: The right triangle formed includes:
- **Hypotenuse**: The radius of the unit circle, which is equal to 1.
- **Vertical Component (Red Line)**: Represents the sine of the angle.
- **Horizontal Component (Blue Line)**: Represents the cosine of the angle.
The task is to determine the exact sine value of the angle \( \frac{5\pi}{3} \) without rounding.
**Solution Steps:**
1. Recall that the unit circle coordinates for an angle \( \theta \) are given by \( (\cos(\theta), \sin(\theta)) \).
2. For angle \( \frac{5\pi}{3} \) radians, which is equivalent to \( -\frac{\pi}{3} \) radians:
- The reference angle in the fourth quadrant is \( \frac{\pi}{3} \).
- The cosine of \( \frac{5\pi}{3} \) is positive, while the sine is negative.
Therefore, \( \
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