Problem 29 - Algebra and Trigonometry

Question

\(25-30\) me measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. $$ -\frac{\pi}{4} $$

Step-by-Step Solution

Answer

Positive: \(\frac{7\pi}{4}, \frac{15\pi}{4}\), Negative: \(-\frac{9\pi}{4}, -\frac{17\pi}{4}\).

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same terminal side when plotted in standard position. Two angles are coterminal if they differ by a multiple of a full circle. In radians, a full circle is represented by \(2\pi\).

Step 2: Find Positive Coterminal Angles

To find a positive angle coterminal with \(-\frac{\pi}{4}\), add \(2\pi\) to the given angle. Calculate:\[-\frac{\pi}{4} + 2\pi = -\frac{\pi}{4} + \frac{8\pi}{4} = \frac{7\pi}{4}\].To find another positive coterminal angle, add another \(2\pi\):\[-\frac{\pi}{4} + 4\pi = -\frac{\pi}{4} + \frac{16\pi}{4} = \frac{15\pi}{4}\].

Step 3: Find Negative Coterminal Angles

To find a negative angle coterminal with \(-\frac{\pi}{4}\), subtract \(2\pi\):\[-\frac{\pi}{4} - 2\pi = -\frac{\pi}{4} - \frac{8\pi}{4} = -\frac{9\pi}{4}\].To find another negative coterminal angle, subtract another \(2\pi\):\[-\frac{\pi}{4} - 4\pi = -\frac{\pi}{4} - \frac{16\pi}{4} = -\frac{17\pi}{4}\].

Key Concepts

Standard Position

When discussing angles, the term "standard position" is a key concept. In mathematics, an angle is said to be in standard position when its vertex is at the origin of a coordinate plane, and its initial side is along the positive x-axis. This is a convention used to make it easier to analyze and compare angles. The other side of the angle is called the terminal side and it rotates either clockwise or counterclockwise from the initial side to form the angle. This setup allows us to easily define and find coterminal angles, which are angles that share the same terminal side. Understanding this concept is crucial as it serves as the foundation for exploring angles in trigonometry.

Radian Measure

Angles can be measured in different units, and radian measure is one of the most commonly used in mathematics, especially in calculus and trigonometry. A radian is the angle subtended at the center of a circle by an arc whose length is equal to the circle's radius. This means that there are precisely \(2\pi\) radians in a full circle, which simplifies many mathematical calculations. For example, in this exercise, we are using radians to find coterminal angles by adding or subtracting multiples of \(2\pi\). Remember, when you encounter a problem involving radians, you can interchangeably convert between radians and degrees for convenience, using the fact that \(180^\circ = \pi\) radians. This understanding helps in solving trigonometric problems effortlessly.

Positive Angles

Positive angles are formed by rotating the terminal side of an angle counterclockwise from the positive x-axis in standard position. When finding positive coterminal angles, we add full circle rotations (\(2\pi\) radians) to the given angle until the angle becomes positive. For the given angle, \(-\frac{\pi}{4}\), adding \(2\pi\) once gives us \(\frac{7\pi}{4}\). Adding it again results in \(\frac{15\pi}{4}\). These new angles not only share the same terminal side as the given angle \(-\frac{\pi}{4}\), but they also lie within the positive range. Visualizing this on a unit circle can help one see how the same terminal side results from different rotations, illustrating the concept of coterminal angles effectively.

Negative Angles

Negative angles are the counterparts of positive angles and are formed by rotating the terminal side clockwise from the positive x-axis. To find negative coterminal angles, the strategy is to subtract full circle rotations (\(2\pi\)) from the given angle. For the angle \(-\frac{\pi}{4}\), when you subtract \(2\pi\) once, you get \(-\frac{9\pi}{4}\), and subtracting it again gives \(-\frac{17\pi}{4}\). These resulting angles, \(-\frac{9\pi}{4}\) and \(-\frac{17\pi}{4}\), are just other forms of describing the same terminal position. It shows how an angle can be represented in multiple ways through negative values, still being coterminal with the original angle \(-\frac{\pi}{4}\). Understanding negative angles is essential for a comprehensive grasp of trigonometry.