Problem 31 - Algebra 2

Question

In which quadrant, or on which axis, does the terminal side of each angle lie? \(\frac{4 \pi}{3}\) radians

Step-by-Step Solution

Answer

The terminal side of the angle \(\frac{4 \pi}{3}\) radians lies in the third quadrant.

Step 1: Understand the value of given angle

The given angle is \(\frac{4 \pi}{3}\). This is greater than \(\pi\), but less than \(\frac{3\pi}{2}\). This indicates that the terminal side will lie in the third quadrant.

Step 2: Make a declaration based on conclusion

Based on the analysis in Step 1, it can be concluded that the terminal side of the angle \(\frac{4 \pi}{3}\) radians lies in the third quadrant.

Key Concepts

Unit Circle

The unit circle is an essential tool in trigonometry. It helps you understand the relationship between angles and their trigonometric values. Imagine a circle with a radius of 1 centered at the origin of a coordinate plane. This is your unit circle.
The circle is divided into four quadrants, with the x-axis and y-axis dieng them into sections. These quadrants help you identify where the terminal side of a given angle lies.
Using the unit circle, you can easily calculate the sine, cosine, and tangent of any angle, as these are represented as coordinates or ratios along the circle's circumference.

Quadrants

Quadrants divide the unit circle into four sections. Each quadrant corresponds to specific angle ranges:

• Quadrant I: angles from 0 to \(\frac{\pi}{2}\)
• Quadrant II: angles from \(\frac{\pi}{2}\) to \(\pi\)
• Quadrant III: angles from \(\pi\) to \(\frac{3\pi}{2}\)
• Quadrant IV: angles from \(\frac{3\pi}{2}\) to 2\(\pi\)

In trigonometry, the position of an angle's terminal side determines its quadrant. For the angle \(\frac{4\pi}{3}\), it falls between \(\pi\) and \(\frac{3\pi}{2}\), which places it in Quadrant III.
Understanding quadrants is key as they also affect the sign (positive or negative) of sine, cosine, and tangent values.

Angles in Radians

Radians offer a way to measure angles based on the radius of the circle. Unlike degrees, which divide a circle into 360 parts, radians measure the angle at the circle's center based on the arc's length. In a full circle, there are 2\(\pi\) radians.
This makes radians a natural way to express angles in mathematics and physics. Each radian corresponds to the angle created when the arc length equals the radius length.
The angle \(\frac{4\pi}{3}\) radians is just over \(\pi\) radians, indicating it is more than half of a circle but less than three-quarters.

Angle Measures

Understanding angle measures in trigonometry involves knowing how to convert between radians and degrees and recognizing common angle values.
For conversions, remember:

• \(\pi\) radians = 180 degrees
• To convert from radians to degrees, multiply by \(\frac{180}{\pi}\)
• To convert from degrees to radians, multiply by \(\frac{\pi}{180}\)

In the exercise, the angle \(\frac{4\pi}{3}\) is presented in radians. To understand it further, conversion into degrees might help. Doing the calculation, \(\frac{4\pi}{3}\) radians equals 240 degrees.
Recognizing these measures helps to easily estimate and locate angles on the unit circle efficiently.