Problem 48 - Algebra 2

Question

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(\frac{\pi}{2}\)

Step-by-Step Solution

Answer

Positive: \(\frac{5\pi}{2}\), Negative: \(-\frac{3\pi}{2}\)

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides, but differ by full rotations. Two angles are coterminal if they differ by a multiple of full rotation (\(2\pi\) radians in this context).

Step 2: Finding a Positive Coterminal Angle

To find a positive coterminal angle of \(\frac{\pi}{2}\), we add \(2\pi\) to \(\frac{\pi}{2}\). So, the calculation is \(\frac{\pi}{2} + 2\pi = \frac{\pi}{2} + \frac{4\pi}{2} = \frac{5\pi}{2}\).

Step 3: Finding a Negative Coterminal Angle

To find a negative coterminal angle of \(\frac{\pi}{2}\), we subtract \(2\pi\) from \(\frac{\pi}{2}\). So, the calculation is \(\frac{\pi}{2} - 2\pi = \frac{\pi}{2} - \frac{4\pi}{2} = -\frac{3\pi}{2}\).

Key Concepts

Angles with Positive Measure

Angles with a positive measure are those measured counterclockwise from their initial side. This gives them a positive value when placed in the context of unit circle and radian measure. To find a coterminal angle with a positive measure, we often add full rotations to the given angle.

• Full rotation in radians is given by \(2\pi\).
• Adding \(2\pi\) to any angle will give a positive coterminal angle.

In the context of the original problem, the angle \(\frac{\pi}{2}\) is given. To find a coterminal angle, you would add \(2\pi\), resulting in an angle of \(\frac{5\pi}{2}\).
This is a straightforward way to obtain angles that are greater than zero which can simplify certain calculations and interpretations, especially when dealing with rotational motion or circular references.
Positive coterminal angles are particularly useful in navigation and periodic phenomena, where understanding the positive cycle of an event is essential.

Angles with Negative Measure

Angles measured clockwise from the initial side have a negative value. Coterminal angles with negative measures are obtained by subtracting multiples of \(2\pi\) from the original angle.

• Subtracting \(2\pi\) shifts the angle in the opposite direction to positive measures.
• This often simplifies calculations that require understanding of opposite direction or reversal of motions.

For example, given the angle \(\frac{\pi}{2}\), you can find a coterminal angle by performing the calculation \(\frac{\pi}{2} - 2\pi = -\frac{3\pi}{2}\).
Using negative coterminal angles can be extremely helpful in scenarios involving rotations or oscillations, where reversal or past cycles need to be understood and calculated.
Such measures are also helpful in fields like signal processing and mechanical rotations where directionality matters.

Full Rotation in Radians

Understanding a full rotation in radians is essential when dealing with coterminal angles. A full rotation is equivalent to \(2\pi\) radians, which means completing one entire circle.

• The radian system is based on the radius of the circle, which aligns seamlessly with the mathematical properties of circles.
• A rotation of \(2\pi\) means the terminal side returns to the starting position, making the angle effectively "repeat."

When solving problems involving angles, the concept of a full rotation aids in understanding how multiple cycles or repetitions can be computed.
Whether you're moving positively or negatively, adding or subtracting \(2\pi\) will not change the original position's terminal side. It simply adjusts the cyclical result.
Mastering this concept makes it easier to compute lengths, areas, and other circular dimensions, often encountered in advanced mathematics and physics applications.