Problem 23 - Precalculus: A Concise Course

Question

Sketch each angle in standard position. (a) \(\frac{\pi}{3}\) (b) \(-\frac{2 \pi}{3}\)

Step-by-Step Solution

Answer

Sketches of \(\frac{\pi}{3}\) and \(-\frac{2\pi}{3}\) in standard position will look like a \(\frac{\pi}{3}\) radians counter-clockwise rotation and a \(\frac{2\pi}{3}\) radians clockwise rotation from the positive x-axis, respectively.

Step 1: Understanding Standard Position

Firstly, it's necessary to understand what it means to sketch an angle in standard position. An angle is said to be in standard position if its vertex is at the origin (0,0) and its initial side is along the positive x-axis.

Step 2: Noting Angle Measurements

The angle \(\frac{\pi}{3}\) is a positive acute angle that lies between 0 and \(\frac{\pi}{2}\). This is found in the first quadrant of the unit circle. The angle \(-\frac{2\pi}{3}\) is a negative acute angle that lies between 0 and \(-\pi\). This is found in the third quadrant of the unit circle.

Step 3: Sketching the Angles

Start by drawing the coordinate plane. Then sketch the angle \(\frac{\pi}{3}\) in standard position by rotating \(\frac{\pi}{3}\) radians counter-clockwise from the positive x-axis. This will result in an angle in the first quadrant. Then sketch the angle \(-\frac{2\pi}{3}\) in standard position by rotating \(\frac{2\pi}{3}\) radians clockwise from the positive x-axis. This results in an angle in the third quadrant.

Key Concepts

Standard Position

The concept of angles in standard position is fundamental when dealing with trigonometry and the unit circle. An angle is in standard position when it has its vertex at the origin of the coordinate plane, specifically at the point (0, 0). Additionally, its initial side lies along the positive x-axis. This framework provides a reference for measuring angles in a consistent manner, which is crucial for understanding rotational movement.

When sketching angles in standard position, it's important to remember:

• The vertex is always at the origin.
• The initial side always points along the positive x-axis.
• Angles are measured from the initial side to the terminal side.
• A positive angle is measured counter-clockwise, while a negative angle is measured clockwise.

These characteristics help standardize how we discuss and sketch angles, providing a uniform baseline for calculations and visualizations.

Unit Circle

The unit circle is an invaluable tool in trigonometry that aids in understanding how angles correspond to real numbers. It is defined as a circle whose radius is exactly one unit. The center of the unit circle is at the origin (0, 0) in the coordinate plane.

Each angle in standard position corresponds to a unique point on the circumference of the unit circle. The angle is measured in radians, where 360 degrees equals 2π radians. This relationship allows us to easily convert between degrees and radians when necessary. For example:

• An angle of 0 radians corresponds to the point (1, 0).
• An angle of 𝜋/2 radians points to (0, 1).
• 3𝜋/2 radians points downwards to (0, -1).
• An angle of 2𝜋 radians takes us back to the starting point, completing the circle.

Understanding these key points on the unit circle helps in the visualization and sketching of angles on the coordinate plane, revealing the symmetry and periodic nature of trigonometric functions.

Coordinate Plane

Sketching angles often takes place in the coordinate plane, which is a two-dimensional space consisting of a horizontal axis (x-axis) and a vertical axis (y-axis). The intersection of these axes is known as the origin, located at (0, 0). The coordinate plane allows us to place and visualize geometric objects and angles systematically.

When dealing with angles in the coordinate plane, a few critical aspects to note are:

• The x-axis is the reference line used for measuring angles in standard position.
• Quadrants divide the plane: the first quadrant contains positive x and y values, the second quadrant contains negative x and positive y values, and so forth, going counter-clockwise.
• The direction of the angle's rotation (counter-clockwise for positive, clockwise for negative) determines its positioning across the coordinate plane quadrants.

The coordinate plane, therefore, serves as the foundation for both constructing and understanding angles, making it an essential tool in mathematics and geometry.