Question 4.1.4: Drawing Angles in Standard Position Draw and label each angl...
Drawing Angles in Standard Position
Draw and label each angle in standard position:
Because we are drawing angles in standard position, each vertex is at the origin and each initial side lies along the positive x-axis.
a. An angle of \frac{\pi}{4} radians is a positive angle. It is obtained by rotating the terminal side counterclockwise. Because 2π is a full-circle revolution, we can express \frac{\pi}{4} as a fractional part of 2π to determine the necessary rotation:
We see that θ =\frac{\pi}{4} is obtained by rotating the terminal side counterclockwise for \frac{1}{8} of a revolution. The angle lies in quadrant I and is shown in Figure 4.11.
b. An angle of \frac{5\pi}{4} radians is a positive angle. It is obtained by rotating the terminal side counterclockwise. Here are two ways to determine the necessary rotation:
Method 1 shows that α=\frac{5\pi}{4} is obtained by rotating the terminal side counterclockwise for \frac{5}{8} of a revolution. Method 2 shows that α=\frac{5\pi}{4} is obtained by rotating the terminal side counterclockwise for half of a revolution followed by a counterclockwise rotation of \frac{1}{8} of a revolution. The angle lies in quadrant III and is shown in Figure 4.12.
c. An angle of -\frac{3\pi}{4} is a negative angle. It is obtained by rotating the terminal side clockwise. We use |-\frac{3\pi}{4}|, or \frac{3\pi}{4}, to determine the necessary rotation.
Method 1 shows that β=-\frac{3\pi}{4} is obtained by rotating the terminal side clockwise for \frac{3}{8} of a revolution. Method 2 shows that β=-\frac{3\pi}{4} is obtained by rotating the terminal side clockwise for \frac{1}{4} of a revolution followed by a clockwise rotation of \frac{1}{8} of a revolution. The angle lies in quadrant III and is shown in Figure 4.13.
d. An angle of \frac{9\pi}{4} radians is a positive angle. It is obtained by rotating the terminal side counterclockwise. Here are two methods to determine the necessary rotation:
Method 1 shows that γ=\frac{9\pi}{4} is obtained by rotating the terminal side counterclockwise for 1\frac{1}{8} revolutions. Method 2 shows that γ=\frac{9\pi}{4} is obtained by rotating the terminal side counterclockwise for a full-circle revolution followed by a counterclockwise rotation of \frac{1}{8} of a revolution. The angle lies in quadrant I and is shown in Figure 4.14.
