Problem 22 - Contemporary Precalculus

Question

Find the radian measure of an angle in standard position that has measure between 0 and \(2 \pi\) and is coterminal with the angle in standard position whose measure is given. $$16 \pi / 3$$

Step-by-Step Solution

Answer

Question: Find the radian measure of an angle in standard position that has a measure between 0 and \(2\pi\) and is coterminal with another angle whose measure is \(\frac{16\pi}{3}\). Answer: \(\frac{4\pi}{3}\)

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same terminal side. In other words, they have the same endpoint when drawn in standard position. We can find coterminal angles by adding or subtracting multiples of full rotations (\(2\pi\) radians) to the given angle.

Step 2: Adding/Subtracting multiples of \(2\pi\) until we have an angle between 0 and \(2\pi\)

Given angle: \( \frac{16\pi}{3}\) We will subtract multiples of \(2\pi\) from the given angle until the resulting angle lies between 0 and \(2\pi\): 1. \(\frac{16\pi}{3} - 2\pi = \frac{16\pi}{3} - \frac{6\pi}{3} = \frac{10\pi}{3}\) Now check if the resulting angle lies between 0 and \(2\pi\): \(\frac{10\pi}{3} > 0\) but \(\frac{10\pi}{3} > 2\pi\) since \(\frac{10}{3} > 2\) So, we will need to subtract another \(2\pi\): 2. \(\frac{10\pi}{3} - 2\pi = \frac{10\pi}{3} - \frac{6\pi}{3} = \frac{4\pi}{3}\) Now check if the resulting angle lies between 0 and \(2\pi\): \(\frac{4\pi}{3} > 0\) and \(\frac{4\pi}{3} < 2\pi\) since \(\frac{4}{3} < 2\)

Step 3: Final Answer

The angle in standard position that has a measure between 0 and \(2\pi\) and is coterminal with the given angle of measure \(\frac{16\pi}{3}\) is \(\boxed{\frac{4\pi}{3}}\).

Key Concepts

Radian Measure

To understand angles and their measurements, one must first grasp the idea of a radian. Unlike the more common degrees, a radian is based on the radius of a circle. A full circle, which is 360 degrees, is equivalent to \(2\pi\) radians. This means that for any given circle, the length of an arc equal to the radius of the circle is one radian.

To convert degrees to radians, multiply by \(\frac{\pi}{180}\). Conversely, to convert from radians to degrees, multiply by \(\frac{180}{\pi}\). This conversion is crucial because many mathematical formulas, including those involving trigonometric functions, use radians as the standard unit of measure for angles.

For example, a 90-degree angle is \(\frac{\pi}{2}\) radians because \((90\times\frac{\pi}{180}=\frac{\pi}{2})\). Using radians can simplify calculations, particularly in calculus and physics, as many natural rates of change are most naturally expressed in radians.

Standard Position Angle

An angle in standard position has its vertex at the origin of a coordinate system, with its initial side lying along the positive x-axis. The angle is measured from the initial side to the terminal side, and can be positive (counter-clockwise direction) or negative (clockwise direction).

When dealing with angles, being able to visualize them in standard position is helpful, especially in trigonometry. The standard position angle helps in determining the quadrant in which the terminal side lies, which is essential when evaluating trigonometric functions. For example, if an angle is in the second quadrant, sine values will be positive, while cosine values will be negative.

Adding and Subtracting Multiples of \(2\pi\)

Angles in a circle can be represented multiple times by adding or subtracting full rotations without changing their terminal position. Since the angle measurement of a complete circle is \(2\pi\) radians, adding or subtracting multiples of \(2\pi\) from any angle will result in an angle that is co-terminal with the original angle.

When simplifying angles to find a co-terminal angle between 0 and \(2\pi\), you subtract (or add) \(2\pi\) until the angle falls within the desired range. This is useful in various fields, including trigonometry and calculus, for finding the primary angle equivalent to a given angle. For instance, if you have an angle of \((\frac{16\pi}{3})\), you can find a co-terminal angle by subtracting \(2\pi\) from it repeatedly until the result is less than \(2\pi\). As seen in the step-by-step solution provided, subtracting \(\frac{6\pi}{3}\) twice gets you \((\frac{4\pi}{3})\), which is between 0 and \(2\pi\) and is, therefore, the co-terminal angle in the specified range.