Problem 50 - Algebra 2

Question

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(\frac{4 \pi}{3}\)

Step-by-Step Solution

Answer

Positive: \(\frac{10\pi}{3}\), Negative: \(-\frac{2\pi}{3}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides when drawn in standard position. To find coterminal angles, we can either add or subtract full circle rotations. A full circle in radians is either added or subtracted by multiples of \(2\pi\).

Step 2: Find the Positive Coterminal Angle

To find a coterminal angle with a positive measure, add \(2\pi\) to \(\frac{4\pi}{3}\): \[\text{Positive Coterminal Angle} = \frac{4\pi}{3} + 2\pi = \frac{4\pi}{3} + \frac{6\pi}{3} = \frac{10\pi}{3}\]

Step 3: Find the Negative Coterminal Angle

To find a coterminal angle with a negative measure, subtract \(2\pi\) from \(\frac{4\pi}{3}\): \[\text{Negative Coterminal Angle} = \frac{4\pi}{3} - 2\pi = \frac{4\pi}{3} - \frac{6\pi}{3} = -\frac{2\pi}{3}\]

Key Concepts

Radian Measure

Radian measure is a method of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians are directly related to the arc length of a circle.
One full rotation around a circle is equal to an angle of \(2\pi\) radians. For a half-circle, the measure would be \(\pi\) radians. Since a radian is based on the radius, it gives a more natural measurement for angles in many mathematical contexts.
Understanding radians is key to solving problems involving coterminal angles and trigonometry in general.

Positive Angle

A positive angle is an angle measured counterclockwise from the positive x-axis. Positive angles are commonly used to represent standard positions of an angle in trigonometry.
To find a coterminal angle with a positive measure, it is usually necessary to add \(2\pi\), or a multiple thereof, to the given angle. This stems from the idea that adding \(2\pi\) results in a complete circle, keeping the terminal side of the angle the same while shifting its measure.

• Example: If the initial angle is \(\frac{4\pi}{3}\), then adding \(2\pi\) yields \(\frac{10\pi}{3}\).

Negative Angle

Negative angles are angles measured clockwise from the positive x-axis. They signify direction but have the same reference points as their positive counterparts.
Finding a coterminal angle in the negative measure usually involves subtraction. By subtracting \(2\pi\) from a given positive angle, the terminal side remains the same, but the angle measure becomes negative.

• Example: Starting with an angle of \(\frac{4\pi}{3}\), subtracting \(2\pi\) gives \(-\frac{2\pi}{3}\).

Angle Subtraction

Angle subtraction plays a vital role in determining negative coterminal angles. When the goal is to find a coterminal angle with a lower (or negative) measure, angle subtraction is the method to use.
Subtracting \(2\pi\) essentially turns the full circle clockwise, resulting in a decrease in the angle's measure.
This operation is straightforward due to its consistent formula: subtract \(2\pi\) from the angle repeatedly until the desired negative measure is achieved.

• Example: For \(\frac{4\pi}{3}\), subtracting \(\frac{6\pi}{3}\) downward shifts the final angle to \(-\frac{2\pi}{3}\).

Angle Addition

In contrast to angle subtraction, angle addition helps in finding a positive coterminal angle by increasing the angle's measure.
Adding \(2\pi\) is akin to moving one full circle forward, which preserves the angular position while increasing the numerical measure. This is particularly useful when dealing with negative angles and transitioning them to positive forms.

• Example: By taking \(\frac{4\pi}{3}\) and adding \(\frac{6\pi}{3}\), the angle measure positively extends to \(\frac{10\pi}{3}\).