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Chapter 1: Problem 3

Draw an angle in standard position of an angle whose radian measure is: (a) \(\frac{1}{4} \pi\) (b) \(\frac{1}{3} \pi\) (c) \(\frac{2}{3} \pi\) (d) \(\frac{5}{4} \pi\) (e) \(-\frac{1}{3} \pi\) (f) 3.4

Short Answer

Expert verified
\((a)\) Angle in standard position with radian measure \(\frac{1}{4}\pi\) is \(45^\circ\) counter-clockwise from the positive x-axis. \((b)\) Angle in standard position with radian measure \(\frac{1}{3}\pi\) is \(60^\circ\) counter-clockwise from the positive x-axis. \((c)\) Angle in standard position with radian measure \(\frac{2}{3}\pi\) is \(120^\circ\) counter-clockwise from the positive x-axis. \((d)\) Angle in standard position with radian measure \(\frac{5}{4}\pi\) is \(225^\circ\) counter-clockwise from the positive x-axis. \((e)\) Angle in standard position with radian measure \(-\frac{1}{3}\pi\) is \(60^\circ\) clockwise from the positive x-axis. \((f)\) Angle in standard position with radian measure 3.4 radians is approximately \(194.8^\circ\) counter-clockwise from the positive x-axis.

Step by step solution

01

(a) Drawing angle: \(\frac{1}{4} \pi\)

Radian measure \(\frac{1}{4} \pi\) corresponds to an angle of \(45^\circ\). To draw this angle in the standard position, place the vertex at the origin (0, 0) and draw the initial side along the positive x-axis. Then, rotate the terminal side counter-clockwise by \(45^\circ\) (which is \(\frac{1}{4}\) of 180^\circ).
02

(b) Drawing angle: \(\frac{1}{3} \pi\)

Radian measure \(\frac{1}{3} \pi\) corresponds to an angle of \(60^\circ\). To draw this angle in the standard position, place the vertex at the origin (0, 0) and draw the initial side along the positive x-axis. Then, rotate the terminal side counter-clockwise by \(60^\circ\).
03

(c) Drawing angle: \(\frac{2}{3} \pi\)

Radian measure \(\frac{2}{3} \pi\) corresponds to an angle of \(120^\circ\). To draw this angle in the standard position, place the vertex at the origin (0, 0) and draw the initial side along the positive x-axis. Then, rotate the terminal side counter-clockwise by \(120^\circ\).
04

(d) Drawing angle: \(\frac{5}{4} \pi\)

Radian measure \(\frac{5}{4} \pi\) corresponds to an angle of \(225^\circ\). To draw this angle in the standard position, place the vertex at the origin (0, 0) and draw the initial side along the positive x-axis. Then, rotate the terminal side counter-clockwise by \(225^\circ\).
05

(e) Drawing angle: \(-\frac{1}{3} \pi\)

Radian measure \(-\frac{1}{3} \pi\) corresponds to an angle of \(-60^\circ\). To draw this angle in the standard position, place the vertex at the origin (0, 0) and draw the initial side along the positive x-axis. Then, rotate the terminal side clockwise by \(60^\circ\), since the angle is negative.
06

(f) Drawing angle: 3.4

To draw an angle with radian measure 3.4, we first need to determine how many degrees this corresponds to. One full rotation around the circle has a radian measure of \(2 \pi\) or approximately 6.283 radians, so 3.4 radians are less than a full rotation. To find the corresponding angle in degrees, we can use the conversion factor \(\frac{180^\circ}{\pi}\). \[3.4\, radians \times \frac{180^\circ}{\pi} \approx 194.8^\circ\] Thus, the angle is approximately 194.8 degrees. To draw this angle in the standard position, place the vertex at the origin (0, 0) and draw the initial side along the positive x-axis. Then, rotate the terminal side counter-clockwise by approximately 194.8 degrees.

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Key Concepts

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

Standard position
In trigonometry, understanding the standard position of an angle is crucial. An angle is said to be in standard position when its vertex is placed at the origin of a coordinate system and the initial side lies along the positive x-axis. This configuration lays the foundation for easily mapping and visualizing angles.
  • The vertex is the starting point (origin).
  • The initial side always aligns with the positive x-axis.
  • The angle's measure, either in degrees or radians, determines the terminal side's position after rotation.
Having angles in standard position simplifies many calculations because the reference framework (x-axis and y-axis) helps in measuring rotational distances accurately.
Angle conversion
Converting between degrees and radians is essential for solving problems in trigonometry and calculus, as both units are widely used. Remember that one full circle is equivalent to 360 degrees or \(2 \pi\) radians. Here鈥檚 how to convert:
  • From degrees to radians: Multiply degrees by \(\frac{\pi}{180^\circ}\).
  • From radians to degrees: Multiply radians by \(\frac{180^\circ}{\pi}\).
For example, if you want to convert \(\frac{1}{4} \pi\) radians to degrees, you multiply it by \(\frac{180^\circ}{\pi}\), giving you 45 degrees. Angle conversion is a vital tool as it allows students to work with the most convenient units for different mathematical contexts.
Counter-clockwise rotation
In mathematics, angles are typically measured by rotating the terminal side of the angle counter-clockwise from the initial side. This is the default direction for positive angle measurements
  • Positive angles rotate counter-clockwise.
  • Negative angles rotate clockwise.
If an angle in standard position is described as \(\frac{1}{3} \pi\) radians, its rotation begins at the positive x-axis and moves counter-clockwise up to its terminal side, here equating to 60 degrees. Contrastingly, an angle with a negative measurement, like \(-\frac{1}{3} \pi\), means you rotate 60 degrees clockwise.
Drawing angles
Visualizing angles involves drawing them in standard positions to grasp their magnitude and orientation better. Here's a step-by-step guide:
  • Start by plotting a point at the origin (0, 0) of your graph, which becomes the vertex of your angle.
  • From this point, draw a straight line along the positive x-axis; this line is your initial side.
  • Rotate about the origin in the counter-clockwise direction for positive angles or clockwise for negative angles.
  • End the rotation once the desired angle measurement is reached, marking your terminal side.
Consider \(3.4\) radians: First, convert this measure to degrees, approximately 194.8, and rotate your terminal side from the positive x-axis counter-clockwise to complete this magnitude. Drawing angles in such a manner helps in visual understanding and problem-solving in geometry and beyond.

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Most popular questions from this chapter

In each of the following, when it is possible, determine the exact measure of central the angle in degrees. Otherwise, round to the nearest hundredth of a degree. (a) The central angle that intercepts an arc of length \(3 \pi\) feet on a circle of radius 5 feet. (b) The central angle that intercepts an arc of length 18 feet on a circle of radius 5 feet. (c) The central angle that intercepts an arc of length 20 meters on a circle of radius 12 meters. (d) The central angle that intercepts an arc of length 5 inches on a circle of radius 5 inches. (e) The central angle that intercepts an arc of length 12 inches on a circle of radius 5 inches.

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