Problem 24 - 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. $$45 \pi / 8$$

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 the given angle \(45 \pi / 8\). Answer: \(\frac{13 \pi}{8}\) radians.

Step 1: Identify the given angle

The given angle is \(\frac{45 \pi}{8}\) radians.

Step 2: Add or subtract multiples of \(2 \pi\) to the given angle

To find a coterminal angle between 0 and \(2 \pi\), add or subtract multiples of \(2 \pi = \frac{16 \pi}{8}\) to the given angle, until the result lies within the specified range.

Step 3: Determine if the given angle can be reduced with addition or subtraction

Since the given angle \(\frac{45 \pi}{8}\) is greater than \(2 \pi = \frac{16 \pi}{8}\), we will need to subtract multiples of \(2 \pi\) from it.

Step 4: Find the smallest coterminal angle within the specified range

Subtract \(\frac{16 \pi}{8}\) from \(\frac{45 \pi}{8}\), once: $$\frac{45 \pi}{8} - \frac{16 \pi}{8} = \frac{29 \pi}{8}$$ Since \(\frac{29 \pi}{8}\) is still larger than \(2 \pi\), we can subtract another multiple of \(2 \pi\): $$\frac{29 \pi}{8} - \frac{16 \pi}{8} = \frac{13 \pi}{8}$$ Now, we have the coterminal angle within the desired range, because \(\frac{13 \pi}{8}\) lies between 0 and \(2 \pi\). The radian measure of the angle in standard position that has a measure between 0 and \(2 \pi\) and is coterminal with the given angle is $$\frac{13 \pi}{8}$$ radians.

Key Concepts

Radian Measure

The radian measure is a standard unit of angular measurement used in mathematics, particularly in trigonometry and calculus. Unlike degrees, a full circle in radian measure is expressed as \(2 \times \text{π} \), making calculations with circular motion and periodic functions much more natural. To understand the concept, imagine the radius of a circle being wrapped along the circle's edge; the length of the arc that corresponds exactly to the radius' length is one radian.

For the exercise, we have an angle of \(\frac{45 \pi}{8}\) radians. To find a coterminal angle, which shares the same starting and ending points but may differ by full rotations, we adjust this angle by subtracting \(\frac{16 \pi}{8} \) (which is equivalent to \(2 \pi\) radians or one full circle) until it falls between 0 and \(2 \times \pi\). Therefore, subtracting once gives us \(\frac{29 \pi}{8}\) and subtracting twice gives us \(\frac{13 \pi}{8}\), a valid radian measure between 0 and \(2 \pi\).

Standard Position

An angle is said to be in standard position when its vertex is located at the origin of a coordinate system and one ray (the initial side) lies along the positive x-axis. The other ray (the terminal side) moves counterclockwise from the initial side for positive angles and clockwise for negative angles.

In terms of our exercise, the angle \(\frac{45 \pi}{8} \) radians is initially not within the 0 to \(2 \pi \) range, so it's not immediately in standard position. By finding the coterminal angle \(\frac{13 \pi}{8}\) that lies within the desired range, we identify the angle's position as it would be measured from the positive x-axis in a standard coordinate system.

Angle Measurement

Angle measurement involves determining the size of an angle and is a fundamental concept in geometry. There are two prevalent systems for measuring angles: degrees and radians. In degrees, a full circle is divided into 360 equal parts, whereas in radians a full circle is \(2 \pi\) radians. One degree is equivalent to approximately \(0.01745\) radians, and one radian is about 57.3 degrees.

Through the context of the problem, we manipulate the given angle's radian measure, \(\frac{45 \pi}{8}\), to align it within the 0 to \(2 \pi\) range. This process is a practical application of angle measurement, showcasing the conversion and calculation techniques. The result, \(\frac{13 \pi}{8}\) radians, is an angle measurement that corresponds with the standard position for angles, illustrating the interconnectedness of these essential trigonometric concepts.