Problem 52 - Algebra 2

Question

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(\frac{9 \pi}{2}\)

Step-by-Step Solution

Answer

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

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position. They can be found by adding or subtracting full rotations of \( 2\pi \) (for radians) or 360° (for degrees) to/from the given angle.

Step 2: Find the Positive Coterminal Angle

To find a coterminal angle with a positive measure for the given angle \( \frac{9\pi}{2} \), we need to subtract \( 2\pi \) until the angle is positive but smaller than a full rotation. Calculate: \[ \frac{9\pi}{2} - 2\pi = \frac{9\pi}{2} - \frac{4\pi}{2} = \frac{5\pi}{2} \] This angle is still greater than \( 2\pi \), so subtract \( 2\pi \) again: \[ \frac{5\pi}{2} - 2\pi = \frac{5\pi}{2} - \frac{4\pi}{2} = \frac{\pi}{2} \] Thus, \( \frac{\pi}{2} \) is a positive coterminal angle.

Step 3: Find the Negative Coterminal Angle

To find a coterminal angle with a negative measure, add \(-2\pi\) to the given angle \(\frac{9\pi}{2}\): \[ \frac{9\pi}{2} - 2\pi = \frac{9\pi}{2} - \frac{4\pi}{2} = \frac{5\pi}{2} \] Now subtract \( 2\pi \) again: \[ \frac{5\pi}{2} - 2\pi = \frac{5\pi}{2} - \frac{4\pi}{2} = \frac{\pi}{2} \] Now add another \(-2\pi\): \[ \frac{\pi}{2} - 2\pi = \frac{\pi}{2} - \frac{4\pi}{2} = -\frac{3\pi}{2} \] So, \(-\frac{3\pi}{2}\) is a negative coterminal angle.

Key Concepts

Angles in Radians

In mathematics, angles can be measured in degrees or radians. Degrees are more common in everyday settings, but radians are frequently used in higher mathematics and science because they offer a direct measure of angles based on the radius of a circle. One full revolution around a circle is equal to \(2\pi\) radians, which corresponds to 360 degrees. This means that \(\pi\) radians equal half a circle, or 180 degrees.

• Radians offer a simple and natural way to express angles, as they directly relate to the properties of circles.
• For example, an angle such as \(\frac{9\pi}{2}\) means traveling 4.5 complete half-circles along the circumference of a circle.

Understanding angles in radians is crucial for solving problems involving periodicity and rotations.

Positive Angles

When we refer to positive angles, these are angles measured in a counter-clockwise direction from the initial side on the positive x-axis. For instance, a positive angle might be one that is larger than 0 radians but smaller than \(2\pi\), such as \(\frac{\pi}{2}\), when considering angles in a circle.

• Positive angles denote the direction of rotation and are essential for determining the position on the coordinate plane.
• In our exploration of coterminal angles, a positive angle equivalent to \(\frac{9\pi}{2}\) is \(\frac{\pi}{2}\).

Getting comfortable with recognizing and applying positive angles is key to mastering algebra and trigonometry concepts.

Negative Angles

Negative angles represent a clockwise direction of rotation starting from the positive x-axis. These angles are less than 0 and appear when you move clockwise around a circle. Given the angle \(\frac{9\pi}{2}\), by repeatedly subtracting \(2\pi\) to reach the smallest positive equivalent, continuing this process will eventually lead to a negative angle.

• An example of a negative coterminal angle for \(\frac{9\pi}{2}\) is \(-\frac{3\pi}{2}\).
• Negative angles are useful in different fields like physics and engineering where direction matters.

Understanding negative angles allows us to fully utilize the rotation properties of angles in various real-world applications.

Angle Rotations

The concept of angle rotations is fundamental when dealing with coterminal angles. Angle rotation describes the complete movement around a circle, equivalent to \(2\pi\) in radians. Rotating an angle either clockwise or counter-clockwise just shifts its position but does not change its terminal side, thereby creating coterminal angles.

• When we say that an angle is rotated by \(2\pi\), we mean it has gone a full circle and returns to the same spot.
• This rotation can generate positive or negative coterminal angles, depending on the direction.

By mastering angle rotations, you gain insight into how angles repeat and overlap in cyclic phenomena, which is an important part of understanding trigonometry.

Algebra 2

Algebra 2 is a critical phase in mathematics education that builds upon the foundation of algebra and introduces more complex concepts, including trigonometrical functions, polynomial equations, and functions. Understanding coterminal angles falls seamlessly under the Algebra 2 curriculum.

• In Algebra 2, students learn how to manipulate algebraic expressions that include angles, both in degrees and radians.
• The study of angles allows learners to explore deeper mathematical concepts and their applications.

Approaching Algebra 2 with a clear understanding of fundamental concepts like angles in radians enhances the capability to solve more advanced mathematical problems.