Drawing Angles in Standard Position and Identifying Quadrants

Q: Draw the angle −π/4 in standard position. State the quadrant in which the angle lies. Choose the correct graph below.

Background

Topic: Angles in Standard Position & Quadrants

This question tests your understanding of how to represent angles in standard position on the coordinate plane and how to determine which quadrant the terminal side of the angle lies in.

Key Terms and Concepts:

• Standard Position: An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis.

• Terminal Side: The ray that rotates from the initial side to form the angle.

• Quadrants: The coordinate plane is divided into four quadrants, labeled I, II, III, and IV, starting from the upper right and moving counterclockwise.

• Negative Angles: Negative angles are measured clockwise from the positive x-axis.

Step-by-Step Guidance

1. Recall that an angle in standard position starts at the positive x-axis. For negative angles, rotate clockwise.

2. Since $-\frac{\pi}{4}$ is negative, rotate $\frac{\pi}{4}$ radians (which is 45°) clockwise from the positive x-axis.

3. Visualize or sketch the terminal side: after rotating 45° clockwise, the terminal side will be in the fourth quadrant.

4. Compare the provided graphs to your sketch. Look for the graph where the terminal side is in the fourth quadrant, making a 45° angle with the x-axis.

Try solving on your own before revealing the answer!

Final Answer:

The correct graph is image_4, and the angle $-\frac{\pi}{4}$ lies in Quadrant IV.

Negative angles are measured clockwise, so $-\frac{\pi}{4}$ places the terminal side in the fourth quadrant.