Problem 48 - Algebra and Trigonometry

Question

\(43-48=\) 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 angle coterminal with \(\frac{51\pi}{2}\) between 0 and \(2\pi\) is \(\frac{3\pi}{2}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides, differing by a full rotation of \(2\pi\) radians. To find an angle coterminal with a given angle, we either add or subtract multiples of \(2\pi\) until the result lies within the desired range.

Step 2: Determine Excess Rotation

Given the angle is \(\frac{51\pi}{2}\). We need to find how many full \(2\pi\) rotations are in \(\frac{51\pi}{2}\) by dividing by \(2\pi\).

Step 3: Performing the Division

Calculate \(\frac{51\pi}{2} \div 2\pi = \frac{51\pi}{2} \times \frac{1}{2\pi} = \frac{51}{4}\). This simplifies to approximately 12.75. Therefore, there are 12 complete rotations, and a partially completed 13th rotation.

Step 4: Finding the Radian Measure

Subtract the 12 full \(2\pi\) rotations from \(\frac{51\pi}{2}\):\[\frac{51\pi}{2} - 12 \times 2\pi = \frac{51\pi}{2} - 24\pi = \frac{51\pi - 48\pi}{2} = \frac{3\pi}{2}\].

Step 5: Verifying the Result

The final result, \(\frac{3\pi}{2}\), is already between 0 and \(2\pi\). Therefore, \(\frac{3\pi}{2}\) is the coterminal angle within the desired range.

Key Concepts

Radian Measure

Radian measure is a way of expressing angles, using the radius of a circle as the unit of measurement. It's different from the more familiar degree system, in which a circle is divided into 360 parts. In radians, one full rotation around a circle is \[ 2\pi \] because the circumference of a circle is \[ 2\pi r, \] where \( r \) is the radius. Here are some important things to remember about radian measure:

• Half of a rotation, or a straight angle, is \( \pi \) radians.
• A right angle is \( \frac{\pi}{2} \) radians.
• Since \( 2\pi \) radians is a full circle, smaller angles are fractions of \( \pi \).

When dealing with coterminal angles and determining angles in the radian system, the task is to find angles within the range from 0 to \( 2\pi \) radians. This is because angles can repeat in cycles, and any angle could have a "twin" repeatedly adding or subtracting \( 2\pi \). Understanding how to work with the radian measure is key when solving problems that involve coterminal angles.

Full Rotation

A full rotation in the context of angles means that an angle has swept through a complete circle and returned to its starting position. In radian measure, a full rotation is expressed as \[ 2\pi \] radians. Whether thinking of this as turning once round a wheel or pivoting completely on one's heel, a full rotation signifies repeating the same path traveled. This idea is important when calculating coterminal angles. Coterminal angles differ by whole multiples of a full rotation:

• If you add \( 2\pi \) to an angle, you complete one additional rotation from where you started.
• Subtracting \( 2\pi \) does the same in reverse, rotating backward once.

In practical terms, when you have an angle like \( \frac{51\pi}{2} \), determining how many full rotations fit into the angle can help simplify it to a more manageable angle that falls within a typical circle's range (0 to \( 2\pi \)). Full rotations are the scaffolding of the problem, helping reveal the simpler path.

Initial and Terminal Sides

In trigonometry, when defining an angle, we talk about two critical parts: the initial side and the terminal side. The initial side is often aligned with the positive x-axis from a standard position. As you draw out an angle, the terminal side is the position it finally reaches as you have rotated around the circle.To visualize this, imagine standing and pointing one arm straight ahead (this is your initial side) and then swinging that arm around like a clock hand (your arm now becomes the terminal side). Let's see how this concept fits into finding coterminal angles:

• If an angle completes a full circle, its terminal side will coincide with the initial side. Even if more full circles are completed, the terminal side remains unchanged.
• Angles coterminal with each other will share the same terminal side, despite starting or ending at different points in the path.

This concept helps in understanding why adding or subtracting complete rotations (\( 2\pi \) in radians) does not change the essential position of the angle's terminal side in relation to its initial side. Knowing this is critical in mathematical problems involving rotations or simplifying large angles.