Problem 33 - Algebra 2

Question

In which quadrant, or on which axis, does the terminal side of each angle lie? \(\frac{9 \pi}{2}\) radians

Step-by-Step Solution

Answer

The terminal side of the angle \(\frac{9\pi}{2}\) lies on the positive y-axis.

Step 1: Convert angle to equivalent within one revolution

The slash in the number suggests that it is a fraction i.e. the angle is \(\frac{9\pi}{2}\). To find the equivalent angle within one revolution, one should divide by \(2\pi\) and check the remainder. Since \(2\pi\) is equal to one complete revolution around a circle, dividing the given angle by \(2\pi\) reveals how many full rotations are made by the angle. The remainder, if any, presents how much of a final rotation is made.

Step 2: Calculate equivalent angle

Upon dividing \(\frac{9\pi}{2}\) by \(2\pi\) one gets \(\frac{9}{4}\). This shows the angle makes 2 full circles (from the 2 in the quotient), with a remainder of \(\frac{1}{4}\). Hence, the reduced angle is \(\frac{1\pi}{2}\).

Step 3: Determine quadrant or axis

An angle of \(\frac{1\pi}{2}\) radians, the equivalent angle, corresponds to an angle of 90 degrees. This angle lies on the positive y-axis. Thus, the terminal side of the angle is on the positive y-axis.

Key Concepts

Radians to Degrees Conversion

Radians and degrees are two units commonly used to measure angles. To understand angles effectively, we often need to convert between these two units. The conversion between radians and degrees is based on the mathematical relationship where the full circle is equal to \(360^\circ\) or \(2\pi\) radians. Because of this, to convert an angle from radians to degrees, we can use the equation:

• Degrees = Radians \( \times \frac{180}{\pi} \)

To see an example, let's convert \(\frac{\pi}{2}\) radians into degrees. Using the conversion formula:

• Degrees = \(\frac{\pi}{2} \times \frac{180}{\pi} = 90^\circ\)

So \(\frac{\pi}{2}\) radians is equivalent to \(90^\circ\). Keep this relationship handy for quick conversions!

Unit Circle

Understanding the unit circle is foundational to trigonometry and converting between radians and degrees. The unit circle is a circle with a radius of one unit, centered at the origin of a two-dimensional coordinate system. This helps in visualizing angles and their trigonometric functions.

• On the unit circle, angles are measured starting from the positive x-axis going counterclockwise.
• The important angles on the unit circle are usually in radians: \(0\), \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), and \(2\pi\).
• These correspond to \(0^\circ\), \(90^\circ\), \(180^\circ\), \(270^\circ\), and \(360^\circ\) respectively.

The beauty of the unit circle is that it provides a visual way to understand which quadrant or axis an angle falls into, as angles move around the circle.

Angle Quadrants

Angles in standard position start from the positive x-axis and move counterclockwise. The coordinate plane is divided into four quadrants:

• Quadrant I: Angles from \(0^\circ\) to \(90^\circ\)
• Quadrant II: Angles from \(90^\circ\) to \(180^\circ\)
• Quadrant III: Angles from \(180^\circ\) to \(270^\circ\)
• Quadrant IV: Angles from \(270^\circ\) to \(360^\circ\)

Besides these quadrants, angles can lie on the axes. For example, an angle of \(\frac{9\pi}{2}\) radians simplifies to \(\frac{\pi}{2}\) radians after accounting for full circles. This is equivalent to \(90^\circ\), which is on the positive y-axis. When analyzing where an angle lies, first convert it to its smallest form by removing multiples of \(2\pi\), then identify the quadrant or axis accordingly.