Problem 68 - Algebra and Trigonometry

Question

Find a positive angle less than \(360^{\circ}\) or \(2 \pi\) that is coterminal with the given angle. $$ -\frac{\pi}{40} $$

Step-by-Step Solution

Answer

The positive angle less than \(360^{\circ}\) or \(2\pi\) that is coterminal with \(-\frac{\pi}{40}\) is \(\frac{79 \pi}{40}\)

Step 1: Understand the given

The given angle is \(-\frac{\pi}{40}\). We need to find a positive angle coterminal to this.

Step 2: Add \(2\pi\) to the angle

Since we are looking for a positive coterminal angle, we add \(2\pi\) to the given angle:\n\(-\frac{\pi}{40} + 2\pi = -\frac{\pi}{40} + \frac{80 \pi}{40}\)

Step 3: Simplify

We simplify the equation obtained in Step 2:\n\(-\frac{\pi}{40} + \frac{80 \pi}{40} = \frac{79 \pi}{40}\)

Key Concepts

Understanding Positive Angles

Angles are a fundamental concept in geometry, helping us understand shapes, patterns, and movement. When we talk about a positive angle, we simply mean an angle that is measured in the counterclockwise direction from the positive x-axis. Imagine the face of a clock; if we start at 12 o'clock and move towards 1 o'clock, we're moving in a positive direction. Just the opposite, if we move from 12 o'clock back towards 11 o'clock, we're moving in a negative direction.

When dealing with angles, it's possible for two angles to share the same starting and ending points; these are called coterminal angles. To find a coterminal angle that is positive, you can add or subtract full rotations of 360 degrees (or in radian measure, \(2\pi\)) until you get an angle in the range of 0 to 360 degrees. This is exactly what the problem at hand illustrates: by adding \(2\pi\) to a negative angle, we can find its positive coterminal counterpart.

Radian Measure Explained

Most students are introduced to angles through degrees, but radians are another essential way to measure angles, especially in calculus and physics. The radian measure is based on the radius of the circle. One radian is the angle created when we wrap the radius of a circle around its circumference. There are \(2\pi\) radians in a full circle, which is equivalent to 360 degrees. This relationship means you can convert between degrees and radians when needed.

To provide a sense of scale, \(\pi\) radians is half a circle, and \(\frac{\pi}{2}\) radians is a quarter-circle, or right angle. The exercise uses the radian measure because it often simplifies calculations in trigonometry, and it's the standard unit in most mathematical functions involving angles.

Angle Simplification

When working with angles, especially in advanced math, angle simplification is a handy tool. Angle simplification involves finding an angle that has the same trigonometric values as the given angle but is within a more manageable range, typically between 0 and \(2\pi\) (or 0 and 360 degrees for degree measure).

This process is beneficial because it allows us to work with smaller, more intuitively understandable values. Simplifying an angle does not change its essential properties, as coterminal angles have the same sine, cosine, and other trigonometric values. The given solution exemplifies angle simplification by adding \(2\pi\) to the negative angle \( -\frac{\pi}{40} \) to arrive at a positive coterminal angle \( \frac{79\pi}{40} \) that is easier to work with and falls within the standard range for positive angles.