Problem 17 - Precalculus: A Concise Course

Question

Determine the quadrant in which each angle lies. (The angle measure is given in radians.) (a) \(\frac{\pi}{4}\) (b) \(\frac{5 \pi}{4}\)

Step-by-Step Solution

Answer

The angle \(\frac{\pi}{4}\) is in the first quadrant and angle \(\frac{5\pi}{4}\) is in the third quadrant.

Step 1: Understanding Quadrants in a Unit Circle

In a unit circle, moving counterclockwise, the first quadrant represents the angles from 0 to \(\frac{\pi}{2}\), the second quadrant denotes angles from \(\frac{\pi}{2}\) to \(\pi\), the third quadrant includes all angles from \(\pi\) to \(\frac{3\pi}{2}\), and the fourth quadrant represents angles from \(\frac{3\pi}{2}\) to \(2\pi\) or \(0\). This directional orientation is important when identifying the quadrant of a given angle.

Step 2: Determining Quadrant for \(\frac{\pi}{4}\)

The given angle \(\frac{\pi}{4}\) lies in the range of 0 to \(\frac{\pi}{2}\), which corresponds to the first quadrant in the unit circle. Hence, the angle \(\frac{\pi}{4}\) is located in the first quadrant.

Step 3: Determining Quadrant for \(\frac{5\pi}{4}\)

The given angle \(\frac{5\pi}{4}\) lies in the range from \(\pi\) to \(\frac{3\pi}{2}\), which corresponds with the third quadrant in the unit circle. Hence, the angle \(\frac{5\pi}{4}\) is located in the third quadrant.

Key Concepts

Understanding Quadrants

The unit circle is an essential tool in trigonometry, offering a way to visually understand angles and their relationships. It's a circle with a radius of 1, divided into four quadrants. Each quadrant is a section of the circle, representing a range of angles. Let's break it down:

• The first quadrant covers angles from 0 to \( \frac{\pi}{2} \)
• The second quadrant holds angles from \( \frac{\pi}{2} \) to \( \pi \)
• The third quadrant includes angles from \( \pi \) to \( \frac{3\pi}{2} \)
• The fourth quadrant spans angles from \( \frac{3\pi}{2} \) back to 0 or \( 2\pi \)

Each quadrant has its own characteristics for functions like sine and cosine, which makes knowing which quadrant an angle lies in very useful. In this exercise, understanding which quadrant an angle is in helps identify the nature and sign of trigonometric values.

Exploring Angle Measures

Angles can be measured in different ways, but in trigonometry, particularly when using the unit circle, they are often measured in radians. Understanding how these angle measures relate to the quadrants is key.

• An angle measure is simply the measure of how much you "rotate" from the starting point, which is typically the positive x-axis.
• When measured in radians, \( \frac{\pi}{4} \) represents a quarter turn within the first quadrant.
• Similarly, \( \frac{5\pi}{4} \) signifies a rotation that lands you in the third quadrant.

For each angle, determine the measure and see where it lands within the circle. This helps you not only pinpoint the quadrant but also predict the behavior of trigonometric functions at that measure.

Significance of Radians

Radians are the standard unit of angular measure used in the context of unit circles. Unlike degrees, which divide a circle into 360 parts, radians give a more mathematical perspective.

• One full circle is \( 2\pi \) radians, going around once.
• A quarter of the circle, or \( \frac{\pi}{2} \), represents ninety degrees.
• This makes radians a more natural fit for equations involving trigonometric functions and calculus.

For example, using radians gives a more straightforward understanding: \( \frac{\pi}{4} \) is an angle smaller than \( \pi \), putting it in the first quadrant, whereas \( \frac{5\pi}{4} \) is beyond \( \pi \), landing in the third quadrant. This unit aligns seamlessly with many mathematical concepts, making it both convenient and efficient in calculations.