Problem 34 - Algebra 2

Question

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

Step-by-Step Solution

Answer

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

Step 1 - Understand the quadrant system

Quadrant I is from 0 to \(\frac{\pi}{2}\) radians, Quadrant II is from \(\frac{\pi}{2}\) to \(\pi\) radians, Quadrant III is from \(\pi\) to \(\frac{3 \pi}{2}\) radians, and Quadrant IV is from \(\frac{3 \pi}{2}\) to \(2 \pi\) radians.

Step 2 - Fall within Quadrant

From the exercise, the given angle is \(\frac{5 \pi}{6}\) radians. According to the quadrant system defined in step 1, \(\frac{5 \pi}{6}\) radians lie between \(\frac{\pi}{2}\) and \(\pi\) radians.

Step 3 - Determine which Quadrant the angle lies in

Since \(\frac{5 \pi}{6}\) radians lie between \(\frac{\pi}{2}\) and \(\pi\) radians, it falls within Quadrant II. Hence the terminal side of \(\frac{5 \pi}{6}\) radians lies in Quadrant II.

Key Concepts

Radians

Radians are a way of measuring angles that offer more precision than degrees. While degrees break a circle into 360 parts, radians divide it based on the radius of the circle itself. A full circle is measured as\[2\pi\] radians. Think of radians as the length of the arc subtended by a central angle in a unit circle, where the radius equals one. This mathematical expression makes calculations cleaner and simpler, especially in trigonometry and calculus.
For angles in radians:

• \( \pi \) radians is equivalent to 180 degrees.
• \( \pi/2 \) radians equals 90 degrees.
• \( 2\pi \) radians complete the circle at 360 degrees.

By understanding this relation, it's possible to easily convert between degrees and radians using the formula:\[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]This relationship is fundamental when navigating through trigonometric problems.

Angle Measurement

Angles are measured in both degrees and radians, but using radians often facilitates easier computation in mathematical equations. Understanding how to switch between these two measurements is crucial. When converting:

• Degrees to Radians, multiply by \( \frac{\pi}{180} \).
• Radians to Degrees, multiply by \( \frac{180}{\pi} \).

This conversion helps in understanding the numerical value of an angle regardless of the measuring unit you're using.
When given an angle like \( \frac{5\pi}{6} \) radians, it's essential to recognize that it's a portion of a circle. Here, it represents a significant part of the half-circle, between \( \frac{\pi}{2} \) (90 degrees) and \( \pi \) (180 degrees).
This translates the angle measurement into a specific position on the circular plane, aiding in more straightforward visual identification of its exact location.

Quadrant System

The Quadrant System is a way to break down a circle into four areas, each representing 90 degrees or its equivalent in radians. Understanding this system helps identify the terminal side of an angle. Each quadrant is defined as:

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

In trigonometric functions, each quadrant has specific characteristics complement to angles and their signs. For instance, sine values are positive in Quadrants I and II, while cosine values are positive in Quadrants I and IV.
Taking an angle like \( \frac{5\pi}{6} \) radians, it falls within Quadrant II based on the defined limits, as it's between \( \frac{\pi}{2} \) and \( \pi \). Understanding which quadrant an angle falls into helps solve trigonometric problems by determining the sign and values of trigonometric functions.