Question

Find the reference angle of each angle. $$ \frac{7 \pi}{4} $$

Short answer

\( \frac{\pi}{4} \)

Step-by-step solution

Understand What a Reference Angle Is

The reference angle is the smallest angle between the terminal side of the given angle and the x-axis. It is always a positive acute angle (between 0 and 90 degrees or 0 and \( \frac{\pi}{2} \) radians).

Determine the Quadrant of the Angle

To find the reference angle for \( \frac{7\pi}{4} \), first determine the quadrant in which this angle is located. Since \( 2\pi = 8\pi/4 \), and \( \frac{7\pi}{4} \) is between \( \pi \) and \( 2\pi \), it is in the fourth quadrant.

Calculate the Reference Angle

In the fourth quadrant, the reference angle is found by subtracting the given angle from \( 2\pi \). Calculate \( 2\pi - \frac{7\pi}{4} \): \[ 2\pi - \frac{7\pi}{4} = \frac{8\pi}{4} - \frac{7\pi}{4} = \frac{\pi}{4} \]

Key concepts

Radian Measure

Angles can be measured in degrees or radians. Radians are a unit of angular measure based on the radius of a circle. One complete revolution around a circle in radians is equal to \(2\pi\) radians, which is equivalent to 360 degrees.
For instance, an angle of \(\frac{\pi}{2}\) radians is 90 degrees, half of the full circle.
To convert degrees to radians, use the formula:
\[ \theta = degrees \times \frac{\pi}{180} \]
Conversely, for radians to degrees:
\[ degrees = \theta \times \frac{180}{\pi} \]
Understanding and mastering radian measure is crucial for angles calculation and trigonometric functions. It creates a basis for more advanced topics like calculus and differential equations.

Quadrants

The coordinate plane is divided into four quadrants, each representing a range of angles.

• First Quadrant (0 to \(\frac{\pi}{2}\)): All trigonometric functions are positive here.
• Second Quadrant (\(\frac{\pi}{2}\) to \(\pi\)): Sine is positive, other functions are negative.
• Third Quadrant (\(\pi\) to \(\frac{3\pi}{2}\)): Tangents are positive, other functions are negative.
• Fourth Quadrant (\(\frac{3\pi}{2}\) to \(2\pi\)): Cosine is positive, other functions are negative.

Knowing the quadrant helps identify the sign of trigonometric functions and the reference angle needed. In the example exercise, \(\frac{7\pi}{4}\) falls in the fourth quadrant.

Angle Calculation

To find the reference angle, follow these steps:

• Identify the given angle and its measure in radians.
• Determine the quadrant in which the angle lies.
• Use the quadrant-specific rule to calculate the reference angle.

In the example, \(\frac{7\pi}{4}\) is our angle. Since it's greater than \(\pi\) but less than \(2\pi\), it resides in the fourth quadrant.
Reference angles are always the smallest angle between the terminal side of the given angle and the x-axis. For the fourth quadrant, subtract the angle from \(2\pi\) to find the reference angle:
\[ 2\pi - \frac{7\pi}{4} = \frac{8\pi}{4} - \frac{7\pi}{4} = \frac{\pi}{4} \]
Hence, \(\frac{\pi}{4}\) is the reference angle of \(\frac{7\pi}{4}\).