Problem 107 - Precalculus

Question

For the following exercises, state the reference angle for the given angle. $$ \frac{-\pi}{8} $$

Step-by-Step Solution

Answer

The reference angle is \( \frac{\pi}{8} \).

Step 1: Understand the Concept of Reference Angle

A reference angle is the smallest positive angle formed by the terminal side of the given angle and the horizontal axis. It is always positive and less than or equal to \( \frac{\pi}{2} \) radians (or \( 90^{\circ} \)).

Step 2: Find the Equivalent Positive Angle

Since the given angle is \( \frac{-\pi}{8} \), we first need to convert this into a positive angle by adding \( 2\pi \). Calculate: \[ 2\pi + \left(-\frac{\pi}{8}\right) = \frac{16\pi}{8} - \frac{\pi}{8} = \frac{15\pi}{8} \] So, \( \frac{15\pi}{8} \) is the equivalent positive angle.

Step 3: Determine the Reference Angle for the Positive Equivalent

The equivalent positive angle \( \frac{15\pi}{8} \) is greater than \( \pi \) but less than \( 2\pi \), so it is in the fourth quadrant. For angles in the fourth quadrant, the reference angle is found using: \[ \text{Reference Angle} = 2\pi - \theta = 2\pi - \frac{15\pi}{8} \] Calculate: \[ 2\pi - \frac{15\pi}{8} = \frac{16\pi}{8} - \frac{15\pi}{8} = \frac{\pi}{8} \] Thus, the reference angle is \( \frac{\pi}{8} \).

Key Concepts

Positive Angle Equivalent

To determine the positive angle equivalent of a negative angle, you simply add a full circle's rotation. A full circle in radians is \( 2\pi \). By doing this, you ensure that the angle is converted into its positive form.
For instance, if the angle given is \( -\frac{\pi}{8} \), adding \( 2\pi \) transforms it as follows:

• Calculate \( 2\pi + (-\frac{\pi}{8}) \).
• This simplifies to \( \frac{16\pi}{8} - \frac{\pi}{8} \).
• The result is \( \frac{15\pi}{8} \).

The angle \( \frac{15\pi}{8} \) is positive and represents the same initial direction as \( -\frac{\pi}{8} \). It's important to understand how to compute the positive angle equivalent as it sets up finding the reference angle.

Reference Angle Computation

The reference angle is a concept used to simplify understanding of angles, especially when trigonometric functions are involved. Essentially, it’s the smallest angle your given angle makes with the x-axis.
Reference angles are always between 0 and \( \frac{\pi}{2} \) radians, always positive.
For an angle depicted in radians, like our equivalent \( \frac{15\pi}{8} \), determining the quadrant is crucial:

• If the angle is greater than \( \pi \) but less than \( 2\pi \), it resides in the fourth quadrant.
• To find the reference angle from a fourth-quadrant angle, subtract it from \( 2\pi \).
• So, \( \text{Reference Angle} = 2\pi - \theta = 2\pi - \frac{15\pi}{8} \).
• This equals \( \frac{16\pi}{8} - \frac{15\pi}{8} = \frac{\pi}{8} \).

The outcome is your reference angle, \( \frac{\pi}{8} \), which helps in understanding the angle's trigonometric behavior.

Angle Conversion to Radians

Converting angles from degrees to radians, or vice versa, is essential in mathematics, especially in calculus and trigonometry.
A circle is \( 360^{\circ} \) and is equivalent to \( 2\pi \) radians. Understanding this proportion helps in conversions:

• Multiply degrees by \( \frac{\pi}{180} \) to get radians.
• For instance, \( 90^{\circ} \) converts to \( 90 \times \frac{\pi}{180} = \frac{\pi}{2} \) radians.

However, in our exercise, we’re mainly dealing with radian measures from the start. Knowing this allows for easy identification of positive equivalents, understanding reference angles, and quadrant locations.
Thus, a strong grasp of radian conversion is beneficial for handling various angle-related tasks seamlessly.