Problem 65 - Precalculus

Question

State in which quadrant or on which axis each angle with the given measure in standard position would lie. $$\frac{13 \pi}{4}$$

Step-by-Step Solution

Answer

The angle \( \frac{13\pi}{4} \) lies in the third quadrant.

Step 1: Understanding Angles in Standard Position

An angle is in standard position if its vertex is at the origin of a coordinate plane and one ray (the initial side) is on the positive x-axis. The measure of an angle determines its position in terms of quadrants or axes.

Step 2: Finding an Equivalent Angle

To determine the position of \( \frac{13\pi}{4} \), first find its equivalent angle within one full rotation (0 to \( 2\pi \)). Since one full rotation is \( 2\pi \), we calculate \( \frac{13\pi}{4} \mod 2\pi \).

Step 3: Calculating Remainder after Division

Convert \( 2\pi \) to a fraction with a denominator of 4: \( 2\pi = \frac{8\pi}{4} \). Divide \( \frac{13\pi}{4} \) by \( 2\pi = \frac{8\pi}{4} \) to find the remainder: \( \frac{13\pi}{4} - 2\pi (1) = \frac{13\pi}{4} - \frac{8\pi}{4} = \frac{5\pi}{4} \).

Step 4: Determining the Quadrant

The equivalent angle is \( \frac{5\pi}{4} \). Since \( \pi = \frac{4\pi}{4} \), and \( \frac{5\pi}{4} > \pi \), it lies in the third quadrant, where angles range from \( \pi \) to \( \frac{3\pi}{2} \).

Key Concepts

Quadrants of the Cartesian Plane

The Cartesian plane is divided into four quadrants which help us to pinpoint where an angle will lie when in standard position. Each quadrant has specific properties based on the signs of the x and y coordinates:

• **First Quadrant:** Both x and y coordinates are positive. Angles in this quadrant range from 0 to \( \frac{\pi}{2}\).
• **Second Quadrant:** The x coordinate is negative, and the y coordinate is positive. Angles here range from \( \frac{\pi}{2} \) to \( \pi \).
• **Third Quadrant:** Both x and y coordinates are negative. Angles stretch from \( \pi \) to \( \frac{3\pi}{2} \).
• **Fourth Quadrant:** The x coordinate is positive, and the y coordinate is negative. Angles in this quadrant range from \( \frac{3\pi}{2} \) to \( 2\pi \).

When working with angles in trigonometry and the Cartesian plane, understanding these quadrants helps to determine the behavior of trigonometric functions and the signs of their values.

Standard Position Angles

An angle is said to be in standard position if its vertex is at the origin of the Cartesian plane, and its initial side is along the positive x-axis. This uniform initial setup allows for a consistent definition of each angle's position with respect to the axes and quadrants.

In standard position, the full circle around the point is \( 2\pi \) radians (360 degrees), making it straightforward to map angles visually. For positive angles, the terminal side (the side opposite the initial side) swings counterclockwise. For negative angles, it goes clockwise.

Knowing these initial parameters simplifies determining what quadrant an angle falls into after rotations. Whether an angle is greater than or less than \(2\pi\), we can always express it in terms of a standard range \([0, 2\pi)\) by reducing it to an equivalent angle.

Angular Equivalence

Angular equivalence is crucial to understanding how angles that may appear different can actually represent the same position around a circle. When we encounter an angle like \( \frac{13\pi}{4} \), it may look complex, but we can reduce it to a more familiar range through subtraction of full circle rotations (\( 2\pi \)).

By calculating \( \frac{13\pi}{4} \mod 2\pi \), we find an equivalent angle within the circle’s 0 to \( 2\pi \) range, which helps make sense of its position. In this example, the division leads us to \( \frac{5\pi}{4} \), which places the original angle in the third quadrant.

Understanding angular equivalence allows mathematicians and students alike to simplify complex angle measures into more manageable parts, aiding in the analysis and visualization of trigonometric functions and other circle-related phenomena.