Problem 48 - Precalculus Mathematics for Calculus

Question

Find an angle between 0 and \(2 \pi\) that is coterminal with the given angle. $$10$$

Step-by-Step Solution

Answer

The angle between 0 and \(2\pi\) that is coterminal with 10 is approximately 3.7168.

Step 1: Identify the Coterminal Angle Formula

Coterminal angles are angles that share the same initial and terminal sides. To find a coterminal angle, you can add or subtract multiples of the full circle, \(2\pi\), to the given angle. The formula is: \ \(\theta_{coterminal} = \theta \pm 2k\pi\,\) \where \(k\) is an integer and \(\theta\) is the given angle.

Step 2: Find the Initial Coterminal Angle

Start by reducing or increasing the given angle, \(10\), to find the first angle \( \theta_{coterminal} \). Subtract \(2\pi\) from 10 to bring it within the desired range: \[\theta_{coterminal} = 10 - 2\pi\approx 10 - 6.2832 = 3.7168.\] The result, \(3.7168\), is already between 0 and \(2\pi\), so it is coterminal with the given angle.

Key Concepts

Radians

Radians are a way of measuring angles. They are based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians use the length of the radius along the circumference. This means a whole circle is equal to \(2\pi\) radians.
One radian is the angle made when you wrap the circle's radius along its edge. This measurement is very useful in mathematics because it relates more naturally to other concepts in trigonometry and calculus. It simplifies many formulas, making them easier to work with.

• Remember: \(\pi\) radians is half a circle (180 degrees).
• \(2\pi\) radians means you've completed a full circle (360 degrees).

When solving problems involving radians, always consider whether you need to convert to degrees or work within radians directly.

Angle Reduction

Angle reduction is a technique used to simplify angles to within a single revolution of a circle. For angles measured in radians, this means bringing them between 0 and \(2\pi\). This is particularly useful in trigonometry when comparing angles or using them in functions.
To "reduce" a given angle like 10 radians, you subtract \(2\pi\) repeatedly until the angle is within the desired range. This is your reduced angle. Here's a reminder of the process:

• Given an angle \(\theta\), subtract \(2\pi\) repeatedly.
• Make sure the result is between 0 and \(2\pi\).

By reducing angles, we simplify the work with trigonometric functions without losing the original angle's attributes.

Trigonometry Concepts

Trigonometry involves the study of angles and how they function on the circle, particularly the unit circle. Understanding coterminal angles is fundamental, as it helps relate different angles that have similar properties.
An angle's coterminality is key when the initial and terminal sides of two angles overlap completely. To find coterminal angles, you use the formula:\[\theta_{coterminal} = \theta \pm 2k\pi\]Where \(k\) can be any integer. This means you can add or subtract full rotations without changing the angle’s position.

• Rotations measured in \(2\pi\) radians.
• Many angles can be coterminal, differing by multiples of \(2\pi\).

Understanding these concepts provides a deeper insight into periodic functions and their applications, especially in physics and engineering.