Problem 19 - Precalculus : Building Concepts and Connections

Question

Sketch the angles in standard position. $$-\frac{9 \pi}{4}$$

Step-by-Step Solution

Answer

First, sketch the paths of two full revolutions around the unit circle in the clockwise direction. Then add an extra quarter of a revolution ending on the negative y-axis. That gives the representation of the angle \(-\frac{9\pi}{4}\) in standard position.

Step 1: Understand the Concept of Negative Angle

A negative angle is drawn in the clockwise direction, opposite to the positive angles which are drawn in the counter-clockwise direction. This means, the rotation for \(-\frac{9\pi}{4}\) will be towards the right.

Step 2: Determine the Number of Revolutions

The angle \(-\frac{9\pi}{4}\) means there are 9 quarters of a pi (or, 9 quarters of a revolution around the circle), which formulates into 2 full revolutions (as \(2\pi\) is one revolution) and an additional quarter revolution.

Step 3: Sketch the Angle

Start sketching from the positive x-axis in the direction of rotation (which in this case is clockwise as the angle is negative). Make two full revolutions around the unit circle. After two full revolutions, sketch another quarter of the revolution. Your terminal side should end on the negative y-axis.

Key Concepts

Angle Measurement

The measurement of angles is a fundamental concept in trigonometry. Angles are typically measured in degrees or radians. In this exercise, the angle is given in radians, specifically as \(-\frac{9\pi}{4}\).

Radians are often used in mathematics because they relate arc length directly to the radius of a circle. One full circle is \(2\pi\) radians, equivalent to 360 degrees. Since \(\pi\) radians equals 180 degrees, \(-\frac{9\pi}{4}\) radian is a negative angle that implies rotation in the opposite direction (clockwise) compared to positive angles.

Understanding angle measurement is crucial because it allows you to determine rotation direction and quantify how far you've rotated around the circle. Often, converting between radians and degrees can be useful to visualize angles more easily.

Revolutions

Revolutions represent the number of complete circles a point makes around a central point. In the context of angles, one revolution equates to an angle of \(2\pi\) radians or 360 degrees.

For the angle \(-\frac{9\pi}{4}\), it is crucial to determine how many full revolutions are made. Breaking it down:

• The total angle is \(-\frac{9\pi}{4}\).
• One revolution equals \(-2\pi\).
• By dividing \(-9\pi/4\) by \(2\pi\), you'll find 2 complete revolutions and an additional quarter revolution (as \(\pi/4\) is \(45\) degrees).

Understanding revolutions helps visualize and simplify complex angles, making it easier to determine the final position of the angle. This process helps in accurately sketching angles on the unit circle.

Unit Circle

The unit circle is a circle with its center at the origin (0, 0) of the coordinate plane and a radius of 1. It is a crucial tool in trigonometry for understanding the relationships between angles and their trigonometric functions.

In this exercise, the unit circle helps us visualize where the terminal side of an angle lies. As the angle \(-\frac{9\pi}{4}\) is measured, it starts from the positive x-axis, moves through full revolutions, and finally stops at the remaining quarter revolution. This ultimately places the terminal side on the negative y-axis.

The utility of the unit circle lies in its simplicity, allowing us to connect angles to their sine, cosine, and tangent values easily. Comprehending movement around the unit circle is essential for anyone solving problems involving angles and trigonometric concepts.

Standard Position

An angle is in standard position when its vertex is at the origin of the coordinate plane and its initial side lies along the positive x-axis.

For \(-\frac{9\pi}{4}\), this means starting at the positive x-axis. Then, because the angle is negative, you move in a clockwise direction. Performing two complete clockwise revolutions (equivalent to \(-4\pi\)) and an additional quarter revolution (\(-\pi/4\)) puts the terminal side on the negative y-axis.

• Initial side: Along the positive x-axis.
• Direction: Clockwise (due to negative angle).
• Revolutions: 2 full, one additional quarter.

Sketching angles in standard position simplifies analysis by providing a consistent starting point, aiding in understanding how angles relate and interact on the coordinate plane.