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Chapter 7: Problem 3

Determine the quadrant where the terminal side of each angle lies. $$\theta=\frac{10 \pi}{3}$$

Short Answer

Expert verified
The terminal side of the angle \( \theta = \frac{10 \pi}{3} \) lies in the third quadrant.

Step by step solution

01

Express the given angle in terms of multiples of \( 2\pi \) or \( \pi \)

The angle \( \theta \) is given as \( \frac{10\pi}{3} \). We should express \( \theta \) in terms of \( 2\pi \) or \( \pi \) to understand how many full rotations it makes around the unit circle and then determine the quadrant that it ends in. For now, \( \theta = \frac{10\pi}{3} = \frac{20\pi}{6} \).
02

Determine the number of full rotations

A full rotation around the unit circle is equivalent to \( 2\pi \). Thus, we compare \( \frac{20\pi}{6} \) with \( 2\pi = \frac{12\pi}{6} \) to find out the number of full rotations. The result is \( 1 \) full rotation (since \( 12\pi / 6 \) is less than \( 20\pi / 6 \)), remaining \( \frac{20\pi}{6} - \frac{12\pi}{6} = \frac{8\pi}{6} = \frac{4\pi}{3} \).
03

Identify the quadrant that \( \theta \) ends in

Now, \( \theta \) is equivalent to \( \frac{4\pi}{3} \), which lies in the interval \( \pi \) to \( \frac{3\pi}{2} \), suggesting it lies in the third quadrant. The angle \( \theta \) finishes its rotation in the third quadrant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Circle
The unit circle is a fundamental concept in trigonometry. It provides a circular framework with a radius of 1, centered at the origin of a coordinate plane.
Since the radius is 1, the unit circle shows how the trigonometric functions work by correlating angles to coordinates along the circle. The angle from the positive x-axis represents a counterclockwise rotation.
  • The circle's circumference is denoted by the angle of \(2\pi\) in radians, representing one full rotation.
  • Sine and cosine values on the unit circle are the x and y coordinates of a point on the circumference corresponding to an angle in standard position.
Understanding the unit circle is crucial to analyzing angles and their positions in different quadrants.
Quadrant Analysis
Quadrant analysis involves determining which of the four quadrants an angle will end in after rotation.
The coordinate plane is divided into four quadrants:
  • Quadrant I: Positive x and y coordinates
  • Quadrant II: Negative x and positive y coordinates
  • Quadrant III: Negative x and y coordinates
  • Quadrant IV: Positive x and negative y coordinates
In the exercise, the goal is to find the terminal quadrant of the angle \(\theta = \frac{10\pi}{3}\).
Initially, we express this angle as \(\frac{20\pi}{6}\) and subtract a full rotation of \(\frac{12\pi}{6}\), leaving \(\frac{4\pi}{3}\). This remaining angle, \(\frac{4\pi}{3}\), falls between \(\pi\) and \(\frac{3\pi}{2}\), placing it in the third quadrant. Thus, quadrant analysis allows us to understand where the terminal side of an angle lies.
Angle Rotation
Angle rotation describes the movement of an angle's terminal side starting from the positive x-axis and moving counterclockwise around the unit circle. This can be thought of as sweeping out a sector of the circle.
  • One full counterclockwise rotation equals \(2\pi\) radians or \(360^\circ\) degrees.
  • An angle can experience multiple full rotations before its terminal quadrant is determined.
For the exercise angle \(\theta = \frac{10\pi}{3}\), it's key first to see it makes more than one full rotation. By subtracting a full \(2\pi \), we find the leftover angle \(\frac{4\pi}{3}\). This leftover angle indicates how far the rotation extends into the third quadrant beyond a full circle.
Understanding angle rotation helps in determining the precise final position of an angle's terminal side.

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