Problem 45 - Algebra and Trigonometry

Question

\(43-48=\) Find an angle between 0 and 2\(\pi\) that is coterminal with the given angle. $$ 87 \pi $$

Step-by-Step Solution

Answer

A coterminal angle is \(\pi\), between 0 and \(2\pi\).

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. To find a coterminal angle between 0 and \(2\pi\), you typically add or subtract multiples of \(2\pi\) from the given angle until it's in the desired range.

Step 2: Determine the Initial Range Problem

The given angle is \(87\pi\). We need to subtract or add \(2\pi\) until the angle is between 0 and \(2\pi\). This translates to finding a remainder when dividing \(87\pi\) by \(2\pi\).

Step 3: Compute the Coterminal Angle

Calculate the coterminal angle by taking \(87\pi\) modulo \(2\pi\). This gives: \(87\pi \mod 2\pi = (87 \mod 2)\pi = 1\pi\) because \(87\div2\) gives a remainder of 1.

Step 4: Verify the Angle

We find that \(1\pi\) is indeed between 0 and \(2\pi\), hence it is the required coterminal angle. The calculation shows that when dividing 87 by 2, the remainder is 1, confirming the angle is \(\pi\).

Key Concepts

Understanding Angle Measurement

In trigonometry and geometry, angle measurement is crucial for understanding various rotations and positions. An angle is formed by two rays originating from a common endpoint called the vertex. The size of an angle is often measured in degrees or radians. This concept is important when dealing with angles on the unit circle, where full rotations are typically expressed in radians.

• Degrees: A full circle is divided into 360 equal parts, making each degree a unit of measure.
• Radians: Here, a full circle is divided into the length of the circle's radius, summing to about 6.28 radians for a complete circle.

Choosing between degrees and radians often depends on the mathematical context, with radians being particularly useful in calculus and higher mathematics. The important takeaway is that different systems help describe the size and rotation level of angles through different expressions.
Understanding the range and measurement of angles helps in finding coterminal angles, which are angles that share the same terminal position on a circle after being measured from the same initial side.

Exploring Modular Arithmetic

Modular arithmetic, sometimes known as 'clock arithmetic,' focuses on the remainder of division instead of the quotient. This method is often used in number theory and cryptography.
For angles, it helps determine coterminal angles like in our example with the angle of \(87\pi\). When looking for a coterminal angle between 0 and \(2\pi\), we use modular arithmetic to "wrap" \(87\pi\) into this desired range.

• Calculating Modulus: In our example, \(87\pi \mod 2\pi\) translates to finding what remains when dividing 87 by 2.
• Result: The leftover, or remainder, is 1, leading us to the coterminal angle \(1\pi\), since \(87 \div 2 = 43\) with a remainder of 1.

Modular arithmetic simplifies finding equivalent angle measurements and helps understand periodic functions and cycles in mathematics, like understanding time on a clock where 24 hours is the same as 0 hours due to the modulo involved.

Working with Radians

Radians offer a different way to measure angles compared to more commonly known degrees. Rather than breaking a circle into 360 parts, radians cut it into lengths that are equivalent to the circle's radius.

• Conversion: One full rotation around a circle is \(2\pi\) radians, which translates to 360 degrees.
• Why use Radians? They simplify equations and expressions in calculus, particularly when dealing with derivatives and integrals of trigonometric functions.

Radians are essential for understanding concepts like coterminal angles, as seen when solving for an angle coterminal with \(87\pi\). The solution required adjusting \(87\pi\) using the full rotation of \(2\pi\) to find an angle within the standard circle. This simplification and direct application show why radians are a preferred unit of measure in mathematical contexts beyond simple geometry, making calculations neat and intuitive.