Problem 60 - Precalculus with Calculus Previews

Question

Find the angle between \(-2 \pi\) and o radians that is coterminal with the angle in Problem 49 . $$ \frac{\pi}{6} $$

Step-by-Step Solution

Answer

The coterminal angle is \(\frac{-11\pi}{6}\).

Step 1: Identify the Given Angle

We are given an angle of \( \frac{\pi}{6} \) and need to find its coterminal angle between \(-2\pi\) and 0 radians.

Step 2: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. They can be found by adding or subtracting multiples of \(2\pi\) from the given angle.

Step 3: Finding a Coterminal Angle

To find a coterminal angle for \( \frac{\pi}{6} \) that lies between \(-2\pi\) and 0 radians, we need to subtract \(2\pi\) (since adding \(2\pi\) would take the angle above 0).

Step 4: Calculate Coterminal Angle

Subtract \(2\pi\) from \( \frac{\pi}{6} \):\[\frac{\pi}{6} - 2\pi = \frac{\pi}{6} - \frac{12\pi}{6} = \frac{-11\pi}{6}\]This results in \( \frac{-11\pi}{6} \), which lies between \(-2\pi\) and 0 radians.

Key Concepts

Understanding Radian Measure

Radian measure is an essential concept in trigonometry. It helps us describe angles in terms of the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians use the circle's own geometry for measurement. In simpler words, radians measure the length of the arc created by the angle in relation to the circle's radius.

• One complete circle is measured as 2π radians, which is analogous to 360 degrees.
• Half a circle, therefore, is π radians, or 180 degrees.

For example, the given angle of \( \frac{\pi}{6} \) means it's one-sixth of a semi-circle (π radians), which is 30 degrees when converted. Radians allow us to work more naturally with trigonometric functions, especially when dealing with periodic behaviors in mathematics and physics.

The Art of Angle Subtraction

When it comes to angle subtraction, it's important to ensure the result is within the desired range. For coterminal angles, which share the same terminal sides, we often need to subtract or add multiples of 2π to keep the angle within specific bounds.
This method works because 2π radians represent a full circle. By subtracting 2π radians from an angle, you effectively rotate the angle back to the same position without changing its terminal side.

• Consider the angle \( \frac{\pi}{6} \) from the problem.
• To bring it within the target range of a more negative range n loop that brings the angle closer.
• Here, subtracting 2π radians ensures the angle's current location ranges between -2π and 0.

Exploring Negative Angles

Negative angles might seem tricky at first, but they're simply angles measured in the opposite direction of positive angles. While positive angles are measured counterclockwise from the initial side, negative angles are measured clockwise.
Understanding negative angles is crucial when working with trigonometric functions or finding angles in specific quadrants. The concept allows for flexibility and a full range of angular movement.

• In our solution, the coterminal angle \( \frac{-11\pi}{6} \) is negative since it's measured clockwise.
• This movement begins from the usual starting position of 0 on the standard circle.
• The placement between -2π and 0 signifies its negative orientation.

Why this matters? It helps with periodic functions, as knowing an angle's sign aids in predicting trigonometric function behaviors and modeling real-world scenarios.