Problem 45 - Precalculus Mathematics for Calculus

Question

Find an angle between 0 and \(2 \pi\) that is coterminal with the given angle. $$\frac{17 \pi}{6}$$

Step-by-Step Solution

Answer

The angle \( \frac{5\pi}{6} \) is coterminal with \( \frac{17\pi}{6} \) and lies between 0 and 2π.

Step 1: Understanding Angle Coterminality

Coterminal angles are angles that differ by a full rotation, which is 2π radians for radian measures. The objective is to find an angle between 0 and 2π that is coterminal with the given angle \(\frac{17\pi}{6}\). This can be done by either adding or subtracting 2π multiples from the original angle.

Step 2: Convert Full Rotations to Sixth Form

Since the original angle is in sixths of π, it helps to express 2π in sixths: \[ 2\pi = \frac{12\pi}{6}. \] This will enable us to subtract it directly from the given angle.

Step 3: Subtract Full Radians

Subtract \( \frac{12\pi}{6} \) from \( \frac{17\pi}{6} \):\[ \frac{17\pi}{6} - \frac{12\pi}{6} = \frac{5\pi}{6}. \] Since \( \frac{5\pi}{6} \) lies between 0 and 2π, it is a valid representative angle in this interval.

Key Concepts

Radians

Radians are a way to measure angles based on the radius of a circle. They are often used in mathematics due to their natural relation to circular geometry.
Unlike degrees, which divide a circle into 360 parts, radians are based on the circle's circumference. One full rotation around a circle is equal to its circumference, which is expressed as \(2\pi\) radians. Thus, \(\pi\) radians represents half of the circle or 180 degrees.

• 1 radian is approximately 57.3 degrees.
• Radians offer a straightforward way to work with trigonometric functions like sine and cosine.

Converting between degrees and radians is crucial in certain mathematical contexts. Remember that converting from degrees to radians requires multiplying by \(\frac{\pi}{180}\). Conversely, converting from radians to degrees means multiplying by \(\frac{180}{\pi}\). Understanding radians helps simplify many mathematical operations involving circular motion or periodic functions.

Angle Conversion

Angle conversion between different units is vital in mathematics and physics. Common conversions include degrees and radians. This is significant because certain problems or contexts require angles in a specific unit.
To convert an angle from degrees to radians, multiply it by \(\frac{\pi}{180}\). For instance, a 90-degree angle is equivalent to \(\frac{\pi}{2}\) radians. Conversely, to convert from radians to degrees, you multiply by \(\frac{180}{\pi}\). Thus, \(\pi\) radians translates to 180 degrees.
In various mathematical settings, knowing how to switch between these units allows for easier computation and understanding. Angle conversion is a fundamental skill that enhances precision and clarity when tackling trigonometric problems, especially those involving periodic functions like sine and cosine.

Full Rotation

A full rotation occurs when an object turns in a complete circle, covering an entire rotation around a point or axis. In radians, this is denoted by \(2\pi\). Understanding full rotations is essential for solving problems involving cyclic or repeating phenomena, such as those found in trigonometry and physics.
Full rotations can be visualized as a 360-degree sweep around a circle. Importantly, angles that differ by one or more full rotations are coterminal. These angles share the same terminal side but might have different initial rotations.
For instance, \(\frac{17\pi}{6}\) is coterminal with \(\frac{5\pi}{6}\) because \(\frac{17\pi}{6} - \frac{12\pi}{6} = \frac{5\pi}{6}\), where \(\frac{12\pi}{6} = 2\pi\) represents the full rotation subtracted to achieve the coterminal angle between 0 and \(2\pi\).
Understanding full rotations helps in recognizing patterns and solving angle problems quickly, especially those that involve periodicity or cyclic behavior.