Problem 7 - Precalculus with Calculus Previews

Question

In Problems \(1-16\), draw the given angle in standard position. Bear in mind that the lack of a degree symbol \(\left(^{\circ}\right)\) in an angular measurement indicates that the angle is measured in radians. $$ -240^{\circ} $$

Step-by-Step Solution

Answer

To draw \\(-240^{\circ}\\), start from the positive x-axis, move 120° counterclockwise past the negative x-axis, and label the terminal side.

Step 1: Identify the Angle Type

Given angle is \(-240^{\circ}\). It has the degree symbol indicating that the angle is measured in degrees.

Step 2: Understand Standard Position

An angle in standard position has its vertex at the origin of the coordinate plane, its initial side along the positive x-axis, and it is measured counterclockwise from the initial side.

Step 3: Determine the Direction of the Angle

Because the angle is \(-240^{\circ}\), it is measured clockwise from the positive x-axis.

Step 4: Draw the Initial Side

Place the initial side along the positive x-axis.

Step 5: Measure the Clockwise Angle

Since the angle is negative, measure \(-240^{\circ}\) by moving clockwise from the initial side. One full rotation clockwise is \(-360^{\circ}\), so \(-240^{\circ}\) is a \(360^{\circ} - 240^{\circ} = 120^{\circ}\) angle counterclockwise from the negative x-axis.

Step 6: Draw the Terminal Side

The terminal side of the angle will be transformed by measuring \(-240^{\circ}\). It will be the same as drawing a \(120^{\circ}\) angle in counterclockwise direction on the negative x-axis, forming a reference triangle in the second quadrant.

Step 7: Label the Angle

Label the angle as \(-240^{\circ}\) on the diagram, indicating its direction and measurement from the positive x-axis.

Key Concepts

Negative Angles

Negative angles can be a little tricky at first, but once you understand the concept, they become easier to visualize and work with. In the context of an angle's measurement, a negative angle suggests a clockwise rotation from the positive x-axis.
Here's how it works:

• When we talk about standard position, the angle starts at the positive x-axis.
• A positive angle is measured counterclockwise, while a negative angle is measured clockwise.
• In the case of \(-240^{\circ}\), we're rotating clockwise from the positive x-axis.

By understanding the direction of rotation indicated by the sign of the angle, you can accurately represent negative angles on the coordinate plane.

Coordinate Plane

The coordinate plane is an essential tool for representing angles, especially when trying to understand their standard position. Imagine a flat, two-dimensional space comprising horizontal and vertical lines intersecting at the center.

• The horizontal line is known as the x-axis.
• The vertical line is called the y-axis.
• These axes intersect at the origin (0,0), which is like the heart of the coordinate plane.
• Angles in standard position have their vertex at the origin, with the initial side along the positive x-axis.

The coordinate plane allows us to visualize the rotation of angles. For a \(-240^{\circ}\) angle, you start from the positive x-axis and rotate backward, placing the terminal side in the second quadrant. Understanding this helps you accurately graph any angle, whether negative or positive.

Degrees to Radians Conversions

When working with angles, it is often necessary to convert between degrees and radians, two units for measuring angles. Understanding this conversion is crucial in trigonometry and calculus.Degrees and radians measure the same thing but in different units:

• Degrees: A circle is divided into 360 equal parts, and each part is one degree.
• Radians: A circle is divided based on the radius's length, where one full rotation around a circle is approximately \(6.2832\) radians, or \(2\pi\) radians.

To convert degrees into radians, you can use the conversion factor \(\pi/180^\circ\). Multiply the degree measure by this factor. For instance, converting \(-240^{\circ}\) to radians involves multiplying: \[-240^{\circ} \times \frac{\pi}{180^{\circ}} = -\frac{4\pi}{3} \text{ radians}\]This relationship helps to solve problems involving trigonometric functions and better understand angles in various mathematical contexts.