Problem 37 - Precalculus : Building Concepts and Connections

Question

Find a positive angle and a negative angle that are coterminal with it. $$\frac{\pi}{6}$$

Step-by-Step Solution

Answer

The positive coterminal angle is \(\frac{13\pi}{6}\) and the negative coterminal angle is \(-\frac{11\pi}{6}\)

Step 1: Find a Positive Coterminal Angle

To find a positive coterminal angle, add \(2\pi\) to the given angle. i.e. \(Positive coterminal angle = \frac{\pi}{6} + 2\pi\)

Step 2: Simplify the Expression

Simplifying the expression yields \(Positive Coterminal Angle = \frac{13\pi}{6}\)

Step 3: Find a Negative Coterminal Angle

To find a negative coterminal angle, subtract \(2\pi\) from the given angle. i.e. \(Negative coterminal angle = \frac{\pi}{6} - 2\pi\)

Step 4: Simplify the Expression

Simplify the expression to get \(Negative Coterminal Angle = - \frac{11\pi}{6}\)

Key Concepts

Positive Coterminal Angle

In trigonometry, angles that share the same terminal side after completing full rotations are called coterminal angles. When seeking a positive coterminal angle, we add one full rotation of 360 degrees or its radian equivalent, which is \(2\pi\), to the given angle.
For an angle \(\frac{\pi}{6}\), we add \(2\pi\) to this angle:

• The given angle: \(\frac{\pi}{6}\)
• Adding a full rotation: \(\frac{\pi}{6} + 2\pi\)

Simplifying this results in \(\frac{13\pi}{6}\). This angle is positive, as it results from adding the full rotation, thus the terminus overlaps with the original angle's position.
Remember, you can create multiple positive coterminal angles by adding \(2\pi\) repeatedly, reflecting the idea that coterminal angles can loop indefinitely positive.

Negative Coterminal Angle

Finding negative coterminal angles involves doing the opposite operation used for positive angles. Here, instead of adding \(2\pi\), we subtract it from the given angle. If we consider the original angle \(\frac{\pi}{6}\), we need to subtract \(2\pi\) to get a negative coterminal angle:

• The given angle: \(\frac{\pi}{6}\)
• Subtracting a full rotation: \(\frac{\pi}{6} - 2\pi\)

When simplified, this calculation results in \(-\frac{11\pi}{6}\).
This angle is negative, meaning it completes its path in the clockwise direction. Just like with positive angles, you can find infinitely many negative coterminal angles by continuing to subtract more full rotations of \(2\pi\), keeping in mind that all these angles terminate at the same point as the original.

Trigonometry

Trigonometry is a branch of mathematics, especially dealing with relationships involving lengths and angles of triangles. It is pivotal in defining periodic functions, deriving features about waves, and mapping angles in cycles. A crucial element is understanding angles and their properties, such as coterminal angles.

Key features of trigonometry include:

• Sine, Cosine, and Tangent: These are the primary trigonometric functions that relate angles with side lengths in right-angled triangles.
• Angular Measure: Angles can be measured in degrees or radians, with \(360^\circ = 2\pi\) radians.
• Periodic Nature: Functions like sine and cosine repeat values in cycles, making coterminal angles relevant for calculations and problem-solving.

Understanding these concepts in trigonometry allows us to effectively work with various periodic phenomena and solve geometric problems. The notion of coterminal angles simplifies the understanding of angles that are seen in cycles, enhancing the insights into rotations and circular movements.