Problem 49 - Algebra and Trigonometry

Question

Find an angle between 0 and 2p that is coterminal with the given angle. $$ \frac{17 \pi}{4} $$

Step-by-Step Solution

Answer

The coterminal angle is \(\frac{\pi}{4}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same terminal side after rotation around the circle. This means adding or subtracting multiples of the full circle (in radians, this is \(2\pi\)) from a given angle.

Step 2: Standard Angle Range

We need to find an angle that lies between 0 and \(2\pi\). This means we require an equivalent angle to \(\frac{17\pi}{4}\) within this range.

Step 3: Subtracting \(2\pi\)

Because \(\frac{17\pi}{4}\) is greater than \(2\pi\), we reduce it by subtracting \(2\pi\) until we find an angle within the desired range. Note that \(2\pi = \frac{8\pi}{4}\) to work with similar denominators.

Step 4: First Subtraction

Subtract \(2\pi\) once: \(\frac{17\pi}{4} - \frac{8\pi}{4} = \frac{9\pi}{4}\). This angle is still greater than \(2\pi\), so we need to subtract again.

Step 5: Second Subtraction

Subtract \(2\pi\) again: \(\frac{9\pi}{4} - \frac{8\pi}{4} = \frac{\pi}{4}\). This angle is now within the range of 0 to \(2\pi\).

Step 6: Final Result

The coterminal angle with \(\frac{17\pi}{4}\) that lies between 0 and \(2\pi\) is \(\frac{\pi}{4}\).

Key Concepts

Coterminal Angles

Coterminal angles are simply angles that share the same terminal side on a circle when drawn in standard position. Imagine rotating around a circle; although the number you started with and ended on may differ, you can end up in the same place on the circle. This is what coterminal angles are all about!
To find a coterminal angle of a given angle, you add or subtract full rotations of the circle. In the world of radians, a full circle is measured as \(2\pi\). Therefore, an angle like \(\frac{17\pi}{4}\) can have coterminal counterparts by adding or subtracting \(2\pi\) repeatedly until you find an equivalent angle within a desired range.
This process helps in places like navigation, physics, and, of course, mathematics, where knowing an equivalent angle in a simpler form makes calculations easier.

Radians

Radians are a way to measure angles without relying on degrees. Instead, they use the radius of a circle to measure how far you've rotated around it. A complete circle is \(2\pi\) radians, which is equivalent to 360 degrees.
Understanding radians is critical because:

• They offer a more natural approach when dealing with trigonometric functions.
• They relate more directly to mathematical formulas used in calculus and higher mathematics.

To convert between radians and degrees, you can use the formula \(\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\). For example, \(\pi\) radians is the same as 180 degrees.

Standard Angle Range

When working with angles in trigonometry, it's often useful to express angles within a standard range. Commonly, this range is from 0 to \(2\pi\) radians for angles in their simplest form.
This is useful because:

• You can more easily compare angles within this range.
• It helps simplify calculations involving trigonometric functions.

In the featured exercise, \(\frac{17\pi}{4}\) was simplified to \(\frac{\pi}{4}\) by repeatedly subtracting \(2\pi\). The angle \(\frac{\pi}{4}\) is equivalent to 45 degrees, which sits comfortably within this standard range, allowing us to know exactly where it lies on the unit circle.