Problem 31 - Precalculus Mathematics for Calculus

Question

The 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

Two positive angles: \(\frac{7\pi}{4}\), \(\frac{15\pi}{4}\). Two negative angles: \(-\frac{9\pi}{4}\), \(-\frac{17\pi}{4}\).

Step 1: Understanding Coterminal Angles

Coterminal angles have the same initial and terminal sides after a full rotation of 360° or in radians, a full rotation of \(2\pi\). This means you can add or subtract \(2\pi\) to/from the given angle to find coterminal angles.

Step 2: Identify Positive Coterminal Angles

Start with the given angle \(-\frac{\pi}{4}\). Add \(2\pi\) to it to find a positive coterminal angle: \(-\frac{\pi}{4} + 2\pi = \frac{7\pi}{4}\). Since \(\frac{7\pi}{4}\) is positive, it is the first positive coterminal angle.

Step 3: Find Second Positive Coterminal Angle

To find another positive coterminal angle, add another \(2\pi\) to the first positive coterminal angle: \(\frac{7\pi}{4} + 2\pi = \frac{7\pi}{4} + \frac{8\pi}{4} = \frac{15\pi}{4}\). Thus, \(\frac{15\pi}{4}\) is the second positive coterminal angle.

Step 4: Identify Negative Coterminal Angles

Start again with \(-\frac{\pi}{4}\) and subtract \(2\pi\) to find a negative coterminal angle: \(-\frac{\pi}{4} - 2\pi = -\frac{\pi}{4} - \frac{8\pi}{4} = -\frac{9\pi}{4}\). This gives the first negative coterminal angle.

Step 5: Find Second Negative Coterminal Angle

Subtract \(2\pi\) once more from \(-\frac{9\pi}{4}\) to find another negative coterminal angle: \(-\frac{9\pi}{4} - 2\pi = -\frac{9\pi}{4} - \frac{8\pi}{4} = -\frac{17\pi}{4}\). Thus, \(-\frac{17\pi}{4}\) is the second negative coterminal angle.

Key Concepts

Angles in Standard Position

Angles in standard position have their vertex at the origin of the coordinate system. The initial side of the angle aligns with the positive x-axis, making it easy to determine the direction of rotation. An angle's terminal side is determined by its rotation from the initial side.
Understanding whether an angle is positive or negative is crucial. A positive angle results from counterclockwise rotation. Conversely, a negative angle results from clockwise rotation. For example, the given angle, \(-\frac{\pi}{4}\), indicates a clockwise rotation from the initial side.

Positive Coterminal Angles

Positive coterminal angles share the same terminal side after completing one or more full counterclockwise rotations. To find these angles, you can add multiples of \(2\pi\) radians, as a full circle is equivalent to \(2\pi\) radians.

• Starting with \(-\frac{\pi}{4}\), add \(2\pi\): \(-\frac{\pi}{4} + 2\pi = \frac{7\pi}{4}\).
• Add another full rotation (\(2\pi\)) to \(\frac{7\pi}{4}\): \(\frac{7\pi}{4} + 2\pi = \frac{15\pi}{4}\).

These calculations provide two positive coterminal angles \(\frac{7\pi}{4}\) and \(\frac{15\pi}{4}\). Each is found by completing further full counterclockwise rotations, maintaining the alignment of the terminal side.

Negative Coterminal Angles

Negative coterminal angles are found by performing full clockwise rotations. This means subtracting multiples of \(2\pi\) radians.

• Starting with \(-\frac{\pi}{4}\), subtract \(2\pi\): \(-\frac{\pi}{4} - 2\pi = -\frac{9\pi}{4}\).
• Subtract another \(2\pi\): \(-\frac{9\pi}{4} - 2\pi = -\frac{17\pi}{4}\).

This process gives two negative coterminal angles, \(-\frac{9\pi}{4}\) and \(-\frac{17\pi}{4}\). These represent further clockwise rotations from the initial position.

Angle Measurement in Radians

The radian is a unit of angle measure based on the radius of a circle. Unlike degrees, radians provide a direct link to the geometry of a circle, making them ideal for mathematical applications, especially in calculus.

• One full circle is \(360^\circ\) or \(2\pi\) radians. This equivalence is essential when working with coterminal angles.
• In the context of the problem, the angle \(-\frac{\pi}{4}\) describes a rotation less than one-quarter of a circle clockwise.

Using radians simplifies the calculation of coterminal angles, as you can easily add or subtract \(2\pi\) to find angles that share the same terminal side. This approach emphasizes the elegance and efficiency of radian measure.