Problem 36 - Algebra 2

Question

In which quadrant, or on which axis, does the terminal side of each angle lie? \(\frac{6 \pi}{5}\) radians

Step-by-Step Solution

Answer

The terminal side of the angle \(\frac{6 \pi}{5}\) radians lies in Quadrant III.

Step 1: Understand the Value of the Angle

The given angle is \(6\pi /5\) radians. Recall that a full rotation in a circle is \(2\pi\) radians, a half circle is \( \pi\) radians, and a quarter circle is \( \pi/2\) radians. Now, \(2\pi\) can be expressed as \(10\pi /5\). \( \pi\) can be expressed as \(5\pi /5\). And \( \pi/2\) can be expressed as \(2.5\pi /5\). So, the given angle \(6\pi /5\) is between \(5\pi /5\) and \(7.5\pi /5\).

Step 2: Associate the Angle Value with Quadrants

Given that a counter-clockwise rotation occurs in the order of quadrants (Quadrant I to Quadrant IV), it can be determined where the terminal side of the given angle lies. A rotation of \(2.5\pi /5\) to \(5\pi /5\) lies in Quadrant I or II. A rotation of \(5\pi /5\) to \(7.5\pi /5\) lies in Quadrant II or III. Hence, as such the given angle \(6\pi /5\) lies somewhere in Quadrant II or III.

Step 3: Identify the Correct Quadrant

Since the angle \(6\pi /5\) is greater than \(5\pi /5\) (180 degrees, or half rotation), but less than \(7.5\pi /5\) (270 degrees, or three-quarter rotation), the terminal side of the angle lies in Quadrant III.

Key Concepts

Radian Measure

Radian measure is a way of expressing angles based on the radius of a circle. Instead of degrees, which divide a circle into 360 parts, radians relate the angle to the circle's radius. One complete rotation around a circle is equal to \(2\pi\) radians. This means:

• \(\pi\) radians is half a circle.
• \(\frac{\pi}{2}\) radians is a quarter of a circle.
• \(3\pi\) radians is one and a half times around the circle.

Understanding radian measure makes it easier to work with angles in calculus and trigonometry. It's a mathematical relationship that ties the geometry of circles to the angles we study.

Angle Rotation

When you think about rotating an angle, it's all about moving around the circle from a starting point. Typically, this starts from the positive x-axis.In trigonometry, angles can rotate in two directions:

• Counter-clockwise rotation is considered positive.
• Clockwise rotation is considered negative.

To determine which quadrant an angle lands in, visualize starting at the positive x-axis and moving counter-clockwise. For example, the angle \(\frac{6\pi}{5}\) radians is a bit more than half a rotation because it’s slightly greater than \(\pi\), placing it in the third quadrant.

Unit Circle

The unit circle is a fundamental tool in trigonometry. It's a circle with a radius of 1, centered at the origin of a coordinate plane. Understanding the unit circle helps solve problems about angles and trigonometric functions.Here’s why it’s helpful:

• The coordinates of points on the unit circle give the sine and cosine of the angle.
• It helps visualize angle rotations in the form of radians.
• Angles starting from the positive x-axis rotate around the circle, passing through the four quadrants.

For the angle \(\frac{6\pi}{5}\), visualizing its position on the unit circle shows it lands in Quadrant III, helping to identify the values of sine and cosine for further calculations.