Problem 11 - Precalculus

Question

For the following exercises, draw an angle in standard position with the given measure. $$ \frac{2 \pi}{3} $$

Step-by-Step Solution

Answer

Draw the angle \( \frac{2\pi}{3} \) counterclockwise from the positive x-axis, landing in the second quadrant.

Step 1: Understand the Angle Measure

The given angle is \( \frac{2\pi}{3} \) radians. Remember that \( 2\pi \) radians is equivalent to a full circle, which is 360 degrees. Therefore, \( \frac{2\pi}{3} \) radians is a fraction of a full circle.

Step 2: Convert Radian to Degree if Needed

Sometimes converting to degrees can help in visualizing the angle. To convert radians to degrees, use the formula: \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \). Applying this:\[ \frac{2\pi}{3} \times \frac{180}{\pi} = 120 \text{ degrees} \]

Step 3: Draw the Angle in Standard Position

To draw an angle in standard position:1. Start with the positive x-axis as the initial side.2. Measure the angle counterclockwise since it is positive.3. Since \( \frac{2\pi}{3} \) radians or 120 degrees is between \( \pi \) (180 degrees) and \( \frac{\pi}{2} \) (90 degrees), the terminal side will be in the second quadrant.4. From the positive x-axis, rotate counterclockwise to form this angle.

Step 4: Verify the Position of the Terminal Side

Ensure that the terminal side correctly aligns with the 120-degree or \( \frac{2\pi}{3} \) radians measurement from the positive x-axis, landing in the second quadrant, to confirm the accuracy of your drawing.

Key Concepts

Radian Measure

Radian measure is a way to express angles using the radius of a circle. One full circle is represented by an angle of \( 2\pi \) radians. This is because the circumference of a unit circle (a circle with radius 1) is \( 2\pi \) times the radius, which makes \( 2\pi \) equal to 360 degrees.
Radian measure is often preferred in mathematics because it simplifies many equations and calculations.

• When you have an angle in radians, you're essentially measuring the length of the arc that subtends from that angle.
• One radian is the angle formed when the arc length is equal to the radius of the circle.

For example, in this exercise, \( \frac{2\pi}{3} \) radians represents an angle that is a fraction of the full \( 2\pi \) radians of a circle. Understanding radian measure helps visualize angles in pure geometric terms rather than relying solely on degrees.

Angle in Standard Position

An angle in standard position is one that is drawn on the coordinate plane starting from a specific point: the positive x-axis. This provides a common starting point for measuring angles and helps in visualizing their positions.
To draw an angle in standard position:

• Place the "initial side" along the positive x-axis.
• Move counterclockwise for positive angles and clockwise for negative angles.
• The "terminal side" is the position of the side after you rotate by the angle's measure.

For the angle \( \frac{2\pi}{3} \), you start from the positive x-axis and rotate counterclockwise through 120 degrees (or \( \frac{2\pi}{3} \) radians). The endpoint, or terminal side of the angle, reveals its placement, which is crucial for further trigonometric calculations or applications.

Quadrants in the Coordinate Plane

The coordinate plane is divided into four quadrants. Quadrants help to identify the location of an angle's terminal side in standard position. These are numbered using Roman numerals:

• Quadrant I: Both x and y coordinates are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive, y is negative.

Each quadrant represents a different sign combination for the coordinates of a point.
For instance, when measuring \( \frac{2\pi}{3} \) radians, you would find its terminal side in Quadrant II due to its counterclockwise rotation from the positive x-axis. Knowing which quadrant an angle lands in helps in determining the trigonometric signs of angles, which is essential in calculating precise values of trigonometric functions.

Degree Conversion

Understanding degree conversion is fundamental when dealing with angles in trigonometry. Degrees and radians are two units for measuring angles.
To convert radians to degrees, multiply the radian measure by \( \frac{180}{\pi} \). This conversion is useful because it enables us to use both systems interchangeably:

• A full circle is \( 360 \degree \) or \( 2\pi \) radians.
• Half a circle is \( 180 \degree \) or \( \pi \) radians.
• Quarter of a circle is \( 90 \degree \) or \( \frac{\pi}{2} \) radians.

In the example, \( \frac{2\pi}{3} \, \text{radians} \times \frac{180}{\pi} = 120 \degree \). This illustrates how the same angle can be represented in both radians and degrees, providing flexibility based on the context or preference in problem-solving.