Problem 50 - Precalculus Mathematics for Calculus

Question

Find an angle between 0 and \(2 \pi\) that is coterminal with the given angle. $$\frac{51 \pi}{2}$$

Step-by-Step Solution

Answer

The coterminal angle is \(\frac{3\pi}{2}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same terminal side. To find a coterminal angle between 0 and \(2 \pi\), we need to find an angle that is equivalent to the given angle modulo \(2 \pi\).

Step 2: Calculate Modulo of Given Angle

First, let's express \(2 \pi\) in fractions with the same denominator as the given angle to make the calculation easier.\[ 2 \pi = \frac{4 \pi}{1} = \frac{8 \pi}{2} \]Now, we need to find \(\frac{51 \pi}{2} \mod \frac{8 \pi}{2}\).

Step 3: Perform the Division

Perform the division of the numerator of \(\frac{51 \pi}{2}\) by \(8\) (the new denominator of \(2 \pi\)):\[ 51 \div 8 = 6.375 \]Since we are interested only in the remainder part, calculate it.

Step 4: Calculate the Remainder

The remainder when dividing 51 by 8 is calculated by the following process:Multiply the integer part of the division by 8: \(6 \times 8 = 48\).Subtract this product from 51:\(51 - 48 = 3\).So, the remainder is 3.

Step 5: Write Down the Coterminal Angle

Using the remainder, we form the coterminal angle:\[ \text{Coterminal angle} = \frac{3 \pi}{2} \]

Step 6: Confirm Coterminality

To confirm, we can check the equation: \[ \frac{51 \pi}{2} = n \cdot 2\pi + \frac{3 \pi}{2} \]Where \(n = 6\), confirming that \(\frac{3 \pi}{2}\) is coterminal with \(\frac{51 \pi}{2}\).

Key Concepts

Angle Measurement

Understanding angles and their measurements is crucial in trigonometry. Angles are geometric figures formed by two rays or line segments with a common endpoint, known as the vertex. We often measure angles in radians or degrees, which helps in various mathematical calculations.

• Degrees: This metric is usually more intuitive for everyday use. A full rotation around a circle equals 360 degrees.
• Radians: In mathematics, especially in trigonometry, radians are preferred. Here, a full circle is equivalent to \(2\pi\) radians. One radian is the angle made when the arc length is equal to the radius.

Radian measurement comes in handy for simplifying complex mathematical operations, like finding coterminal angles, which brings a clear sense of the circle's geometry into the calculations. In our problem, first transforming the angle, \(\frac{51 \pi}{2}\), into a form compatible with \(2\pi\) is an initial step towards calculation. Recognizing when your angles exceed the typical range, such as 0 to \(2\pi\), is essential here.

Modulo Operation

The modulo operation is a mathematical concept that helps find the remainder of a division operation. It's especially useful in angle calculations to bring angles to a standard interval, like converting angles to be within \([0, 2\pi)\).
When we perform the modulo operation in trigonometry, we are interested in finding angles that are equivalent within one full rotation \(2\pi\). Here's how it helps:

• Finding equivalence: Modulo lets us reduce larger angles to fall into the standard range that trigonometric functions naturally work with.
• Simplifying calculations: By focusing only on the remainder, we ignore full rotations, which do not affect trigonometric results.

In the given exercise, \(\frac{51 \pi}{2}\) is a multiple of \(2\pi\), with some leftover angle. Finding this leftover using modulo is crucial to identify the precise angle in typical trigonometric functions.

Trigonometric Concepts

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, and functions that describe cyclical phenomena. One important aspect is understanding coterminal angles, which occur when angles have the same initial and terminal sides.
Working with coterminal angles, we look at angles differing by whole rotations — multiples of \(2\pi\) radians — such that they essentially "land" at the same position on a unit circle.

• Applications: Trigonometric functions repeat, or "cycle," their values every \(2\pi\). Thus, angles like \(\frac{3 \pi}{2}\) and \(\frac{51 \pi}{2}\) yield the same trigonometric results.
• Relevance: Recognizing coterminal angles is essential for simplifying complex equations and ensuring angles fit within expected ranges.

Understanding this allows us to calculate more complex phenomena, predict periodic behaviors, and solve applied problems involving waves, oscillations, and circular motion. In this exercise, understanding that \(\frac{3 \pi}{2}\) is coterminal with \(\frac{51 \pi}{2}\) simplifies how one approaches these types of problems.