Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine the quadrant in which each angle lies. (The angle measure is given in radians.) (a) \(-\frac{5 \pi}{12}\) (b) \(-\frac{13 \pi}{9}\)

Step-by-Step Solution

Answer

The angle \(a = -\frac{5 \pi}{12}\) lies in the 3rd quadrant and the angle \(b = -\frac{13 \pi}{9}\) lies in the 2nd quadrant.

Step 1: Determining the Quadrant for Angle (a)

Given angle (a) is \(-\frac{5 \pi}{12}\). Add \(2\pi\) to make the angle positive: \(-\frac{5 \pi}{12} + 2\pi = -\frac{5 \pi}{12} + \frac{24 \pi}{12} = \frac{19 \pi}{12}\). The angle \( \frac{19 \pi}{12} \) lies in the 3rd quadrant because it’s more than \( \pi \) but less than \( \frac{3\pi}{2} \).

Step 2: Determining the Quadrant for Angle (b)

Given angle (b) is \(-\frac{13 \pi}{9}\). Add \(2\pi\) to make the angle positive: \(-\frac{13 \pi}{9} + 2\pi = -\frac{13 \pi}{9} + \frac{18 \pi}{9} = \frac{5\pi}{9}\). The angle \( \frac{5\pi}{9} \) lies in the 2nd quadrant because it’s more than \( \frac{\pi}{2} \) but less than \( \pi \).

Key Concepts

Angles in Radians

When working with angles in trigonometry, you'll often encounter radians, a unit of measurement that is preferred in mathematical calculations. Unlike degrees, where a full circle is 360 degrees, a full circle in terms of radians is represented by the value of \(2\pi\). This means that a radian is directly related to the circle's circumference. To get a sense of scale:

• \(\pi\) radians is equivalent to 180 degrees.
• \(\pi/2\) radians represents a right angle, or 90 degrees.

Using radians can simplify the process when finding arc lengths and areas of sectors in circles, making it a crucial concept in calculus and advanced mathematics. Understanding this conversion between degrees and radians is essential for solving many trigonometric problems.

Positive Angle Conversion

Sometimes, angles are presented in negative form, especially when dealing with scenarios that involve rotating in reverse direction or clockwise around the circle. To simplify trigonometric calculations, we often convert these negative angles into positive angles by performing a simple addition.
To convert a negative angle to its positive counterpart:

• Add \(2\pi\) to the negative angle. This is similar to completing a full circle and aligns the angle to an equivalent positive measure.
• For instance, a negative angle like \(-\frac{5\pi}{12}\) is transformed by adding \(2\pi\), resulting in a positive angle \(\frac{19\pi}{12}\).

By doing this, it becomes easier to identify the quadrant where the angle resides, and apply standard trigonometric functions.

Quadrant Identification

Determining which quadrant an angle falls into is fundamental in trigonometry. The circle is split into four quadrants:

• 1st Quadrant: From 0 to \(\frac{\pi}{2}\) radians. Angles here are both positive and acute.
• 2nd Quadrant: From \(\frac{\pi}{2}\) to \(\pi\) radians. Here, sine values remain positive while the cosine becomes negative.
• 3rd Quadrant: From \(\pi\) to \(\frac{3\pi}{2}\) radians. Both sine and cosine have negative values.
• 4th Quadrant: From \(\frac{3\pi}{2}\) to \(2\pi\) radians. Sine values are negative and cosine values are positive.

After converting your negative angle into a positive one, check which quadrant it falls into by comparing it against these radian benchmarks. For example, \(\frac{5\pi}{9}\) falls in the 2nd quadrant because it is more than \(\frac{\pi}{2}\) but less than \(\pi\), which can help in understanding the behavior of various trigonometric functions at that angle.