Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Find the angle of least positive measure that is co terminal with the given angle. $$-\frac{\pi}{4}$$

Step-by-Step Solution

Answer

The least positive coterminal angle is \(\frac{7\pi}{4}\).

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides when drawn in standard position. To find a coterminal angle, you can add or subtract multiples of a full circle (in radians, that would be \(2\pi\)).

Step 2: Add a Full Rotation

Since \(-\frac{\pi}{4}\) is a negative angle, we want to find a positive angle that is coterminal with it. Add \(2\pi\) to \(-\frac{\pi}{4}\) to find a coterminal angle:\[-\frac{\pi}{4} + 2\pi = -\frac{\pi}{4} + \frac{8\pi}{4} = \frac{7\pi}{4}\]

Step 3: Verify the Angle

The computed angle \(\frac{7\pi}{4}\) is positive and falls within the range \([0, 2\pi)\). Therefore, it is a valid coterminal angle for \(-\frac{\pi}{4}\).

Key Concepts

Radians

Understanding radians is key when working with angles, especially in trigonometry. A radian measures angles based on the circumference of a circle.
Imagine a circle with a radius of 1. If you take the radius length and "wrap" it along the circle's edge, the angle formed at the circle's center measures exactly one radian.

• A full circle is approximately 6.28 radians, or exactly \(2\pi\) radians.
• The radian system is naturally paired with the properties of a circle, making it popular in mathematics.
• There are 360 degrees in a circle, so \(180\) degrees is equivalent to \(\pi\) radians.

When dealing with coterminal angles, radians allow for simple addition and subtraction of \(2\pi\), or full circle rotations, to find equivalent angles. Since these angles repeat every \(2\pi\) radians, we can use this to find positive or negative coterminal angles easily.

Standard Position

Placing an angle in standard position helps simplify the study of angles and their relations.

In standard position, an angle’s initial side lies along the positive x-axis of a coordinate plane. The vertex of the angle remains at the origin, and the angle measures counterclockwise from the initial side to the terminal side.

• An angle in standard position is easy to visualize and manipulate mathematically.
• It allows consistent comparison between different angles, as they share a common starting point.
• Being in a coordinate plane also helps relate angles directly with trigonometric functions.

This approach makes determining coterminal angles more straightforward, as any rotations by \(2\pi\) radians will result in angles that share the same terminal side. Thus, any angle expressed in this manner can be easily converted back and forth from degrees to radians.

Full Circle Rotation

A full circle rotation is the movement around a circle until you return to your starting point, representing a complete loop.

In terms of angles, this is quantified as \(360\) degrees or \(2\pi\) radians. This full rotation concept is pivotal when discussing coterminal angles.

• Adding \(2\pi\) radians to an angle returns you to the same directional orientation.
• This allows one to calculate positive or negative coterminal angles by simply adding or subtracting full circle rotations.
• It shows how any angle greater than \(360\) degrees (or \(2\pi\) radians) has at least one coterminal angle less than \(360\) degrees.

The beauty of full circle rotations is their repetitive nature, aiding in a better understanding of angles and creating an infinite set of angles sharing the same terminal side. Hence, when given a negative angle like \(-\frac{\pi}{4}\), adding \(2\pi\) shifts it into its positive coterminal angle, \(\frac{7\pi}{4}\).