Problem 47 - Algebra and Trigonometry

Question

Find an angle between 0 and 2p that is coterminal with the given angle. $$87 \pi$$

Step-by-Step Solution

Answer

The coterminal angle is \(\pi\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. To find coterminal angles, we can add or subtract multiples of \(2\pi\) from the given angle. This helps us find equivalent angles within the desired range.

Step 2: Determine the Nearest Integer Multiple of \(2\pi\)

First, identify how many full rotations (in terms of \(2\pi\)) are within \(87\pi\). Since a full rotation is \(2\pi\), we divide the given angle by \(2\pi\): \[\frac{87\pi}{2\pi} = 43.5\].

Step 3: Adjust the Angle to Be Between 0 and \(2\pi\)

The quotient from Step 2 is 43.5, indicating 43 complete rotations plus an extra half rotation. We take the integer part (43) and multiply it by \(2\pi\): \[43 \times 2\pi = 86\pi\]. Subtract this result from the original angle to find the coterminal angle within a single rotation: \[87\pi - 86\pi = \pi\].

Step 4: Verify the Coterminal Angle

Verify that the angle \(\pi\) is between 0 and \(2\pi\). Since \(\pi\) is 3.14 and \(2\pi\) is approximately 6.28, the angle \(\pi\) falls within our desired range.

Key Concepts

Angle Measurement

Understanding angle measurement is essential, especially when dealing with coterminal angles. Angles are commonly measured in radians or degrees. Radians are particularly useful in mathematics because they tie directly to the arc length of a circle. In our problem, the angle is given in radians.

• The total circumference of a unit circle is measured as \(2\pi\) radians.
• One full rotation around the circle is \(360^{\circ}\), equivalent to \(2\pi\) radians.

Angles can be larger or smaller than \(2\pi\) radians. By adding or subtracting \(2\pi\) from an angle, we find angles that are 'coterminal,' meaning they share the same starting and ending points on the circle. This can help us identify an equivalent angle within a standard range, specifically between 0 and \(2\pi\). This is crucial in simplifying problems or finding solutions within defined limits.

Full Rotation

A full rotation refers to turning around a complete circle one time. In terms of angle measurement, a full rotation equates to \(360^{\circ}\) or \(2\pi\) radians. Our original angle of \(87\pi\) represents multiple rotations.

• To find out how many full rotations are contained within \(87\pi\), we divide by \(2\pi\).
• This operation yields \(43.5\), indicating 43 complete rotations and an extra half rotation.

Subtracting \(43\) full rotations (i.e., \(86\pi\)), we simplify the issue, leaving a simpler angle of \(\pi\). Such calculations help to determine where an angle lands relative to a start point after a large movement, making complex angles easier to handle.

Unit Circle

The unit circle is a fundamental concept in trigonometry, forming a circle with a radius of one. This setting allows for a straightforward connection between angles and coordinates.

• A point on the unit circle corresponds to an angle scaled from \(0\) to \(2\pi\).
• A full revolution takes us back to the same spot, thus illustrating how coterminal angles work within the circle.

In the scenario where the angle reduces to \(\pi\), it means the point on the unit circle corresponds to this specific angle, aligning to the negative x-axis. This graphical understanding aids in visualizing the rotation and final position of angles, allowing us to determine the coterminal angle accurately. It's a helpful visualization for linking theoretical math with tangible outcomes, especially when grappling with large angles like \(87\pi\).