Problem 5 - 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. $$ 1140^{\circ} $$

Step-by-Step Solution

Answer

The angle \(1140^{\circ}\) in standard position is equivalent to \(60^{\circ}\).

Step 1: Identify Conversion

The given angle is in degrees: \(1140^{\circ}\). Since we need to draw it in standard position, we need to determine how many full rotations (each being \(360^{\circ}\)) are contained in \(1140^{\circ}\).

Step 2: Calculate Full Rotations

Divide \(1140\) by \(360\) to find the number of complete rotations: \(\frac{1140}{360} = 3.1667\). This means the angle completes 3 full rotations and has a remainder.

Step 3: Find Remainder Angle

Calculate the remainder by subtracting the total angle covered by full rotations (\(3 \times 360 = 1080\)) from the original angle: \(1140 - 1080 = 60^{\circ}\).

Step 4: Draw the Angle in Standard Position

Draw a coordinate plane. Starting from the positive x-axis, go counterclockwise to represent the number of full rotations plus the remainder. Visualize the final position after three full circles, then finishing at \(60^{\circ}\) into the fourth rotation.

Key Concepts

Radian and Degree Conversion

Angles can be expressed in degrees or radians, and often it's necessary to convert between the two. One full rotation around a circle is equivalent to 360 degrees or \(2\pi\) radians. To convert degrees into radians, multiply the number of degrees by \(\frac{\pi}{180}\). For example, to convert \(1140^{\circ}\) into radians, you compute:

• \(1140 \, \text{degrees} \times \frac{\pi}{180} = 19.89 \, \text{radians}\)

Conversely, to convert radians to degrees you would multiply by \(\frac{180}{\pi}\). This knowledge is essential because some scenarios might present angles without a degree symbol, indicating the measurement is in radians.

Full Rotations

Understanding full rotations helps in visualizing angles above 360 degrees or their radian equivalent. A full rotation equals 360 degrees, making it vital to determine how many complete rotations an angle covers if it's more than one complete circle.

• In the problem with \(1140^{\circ}\), making full rotations involves dividing by \(360\).
• The result \(3.1667\) indicates 3 full circles have been completed.

Knowing how many full rotations have occurred helps in drawing such angles in their correct ending position along the coordinate plane.

Coordinate Plane

The coordinate plane is a vital part of drawing angles in standard position. It consists of two perpendicular axes: a horizontal x-axis and a vertical y-axis. Standard position always starts from the positive x-axis, moving counterclockwise to measure positive angles.

• To draw \(1140^{\circ}\), begin from the positive x-axis.
• Complete the full rotations, going around the plane completely 3 times.
• Finally, progress 60 degrees further, as the remainder after the rotations constitutes the final angle in standard position.

Visualizing and plotting on the coordinate plane helps in better understanding angles, full rotations, and their terminal sides.

Counterclockwise Measurement

Angles in standard position are measured in a counterclockwise direction from the positive x-axis. This method is standard in mathematics and helps maintain consistency when interpreting trigonometric functions and rotational motion.

• Positive angles move counterclockwise, while negative angles move clockwise.
• The counterclockwise direction aligns with the increase in angle measurement.

In the case of the \(1140^{\circ}\) angle, you would start at the positive x-axis and rotate counterclockwise through complete circles, ending 60 degrees into the next rotation. This helps ensure an accurate representation of the angle's measurement.