Problem 64 - A Graphical Approach to Precalculus with Limits

Question

Give an expression that generates all angles coterminal with each angle. Let \(n\) represent any integer. $$-\frac{7 \pi}{6}$$

Step-by-Step Solution

Answer

The expression is \(\frac{-7\pi + 12\pi n}{6}\), where \(n\) is any integer.

Step 1: Understanding Coterminal Angles

To find coterminal angles, you add or subtract full rotations. In radians, one full rotation equals \(2\pi\). So, coterminal angles of an angle \(\theta\) can be expressed as \(\theta + 2\pi n\), where \(n\) is any integer.

Step 2: Identify the Initial Angle

The angle given in the problem is \(-\frac{7\pi}{6}\). This angle will serve as \(\theta\) in our general expression for coterminal angles.

Step 3: Formulate Expression for Coterminal Angles

Using the formula \(\theta + 2\pi n\), substitute \(\theta = -\frac{7\pi}{6}\). This gives us the expression for coterminal angles: \[-\frac{7\pi}{6} + 2\pi n\]

Step 4: Simplify the Equation

Combine the terms in the expression:\[-\frac{7\pi}{6} + \frac{12\pi n}{6} = \frac{-7\pi + 12\pi n}{6}\]This represents all angles that are coterminal with \(-\frac{7\pi}{6}\).

Key Concepts

Radians

Radians are a unit of angular measure used in many areas of mathematics. Unlike degrees, which divide a circle into 360 parts, radians relate an angle to the radius of the circle. The circumference of a circle is represented as a full rotation, which is \(2\pi\) radians. Thus, an angle measuring \(2\pi\) radians encloses the entire circle. For example, half a rotation or a semi-circle is \(\pi\) radians, while a quarter turn equates to \(\frac{\pi}{2}\) radians. Radians help in precise mathematical calculations because they connect directly to the concept of a circle. This unit aids in determining coterminal angles, as seen in the calculation of \(-\frac{7\pi}{6}\) with additional rotations.

Full Rotation

A full rotation in the realm of angles, whether in radians or degrees, signifies a complete circle around a point. It measures \(2\pi\) radians, equivalent to 360 degrees. This equivalence is foundational when deciphering variations of angles that still point the same direction, known as coterminal angles. To find coterminal angles with any given angle, you add or subtract full rotations. This is because every \(2\pi\) radians covers the entire circle and results in overlapping angles.

• Full Rotation in Degrees: 360°
• Full Rotation in Radians: \(2\pi\)

When considering \(-\frac{7\pi}{6}\), by adding or subtracting multiples of \(2\pi\) (full rotations), we derive all angles coterminal to the original angle.

General Expression

The general expression for finding all coterminal angles is paramount in trigonometry. This expression uses the angle \(\theta\) and full circle multiples to depict every angle sharing the same terminal side. In formula terms, for an angle \(\theta\), coterminal angles are expressed as \(\theta + 2\pi n\), where \(n\) can be any integer. This shows that by adding full rotations \(2\pi\), positive (addition) or negative (subtraction) of any magnitude, we maintain an equivalent angle position. For example, given the angle \(-\frac{7\pi}{6}\), substituting into the general expression yields:

• Expression: \(-\frac{7\pi}{6} + 2\pi n\)
• This form allows calculation of infinite coterminal angles, \(n\) being any integer.

By simplifying, you ensure clarity in examples and universal application.