Problem 40 - Algebra 2

Question

Sketch each angle. Then find its reference angle. \(\frac{13 \pi}{7}\)

Step-by-Step Solution

Answer

The reference angle for \( \frac{13\pi}{7} \) is \( \frac{6\pi}{7} \).

Step 1: Understanding the Terminology

Before we solve the problem, it's important to understand that a reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis. It is always positive and less than or equal to \( \pi \) radian (180 degrees).

Step 2: Determine the Basic Circle Representation

Given the angle \( \frac{13\pi}{7} \), we need to first determine which quadrant this angle lies in. Since \( 2\pi \) is equivalent to a full circle (or 1 full rotation, which is roughly 6.28 radians), we need to evaluate the position of \( \frac{13\pi}{7} \).

Step 3: Calculate Revolutions in Turns of \(2\pi\)

Compute how many full circles \( \frac{13\pi}{7} \) makes: \( \frac{13\pi}{7} \div 2\pi = \frac{13}{14} \), which is less than 1, indicating it's within the first rotation. However, it's greater than \( \pi \) (which is \( \frac{7\pi}{7} \)), indicating it's beyond halfway but less than \( 2\pi \).

Step 4: Determine Quadrant Location

Since \( \frac{13\pi}{7} \) is greater than \( \pi \) but less than \( 2\pi \) \((about 1.86\pi)\), it falls into the third quadrant. A value between \( \pi \) and \( \frac{3\pi}{2} \) ensures the angle lies in this specific location.

Step 5: Calculate Reference Angle

The formula for the reference angle when the angle is in the third quadrant is: \( \text{Reference angle} = \text{Angle} - \pi \). Therefore, \( \text{Reference angle} = \frac{13\pi}{7} - \pi = \frac{13\pi}{7} - \frac{7\pi}{7} = \frac{6\pi}{7} \).

Step 6: Sketch the Angle

Draw the x-y plane. Mark the initial side on the positive x-axis. Since \( \frac{13\pi}{7} \) is in the third quadrant, rotate the terminal side from 0, past \( \pi \) (or 180 degrees), stopping at the third quadrant to an angle equivalent to \( \frac{13\pi}{7} \). Finish by noting the reference angle, \( \frac{6\pi}{7} \).

Key Concepts

Quadrants

Understanding quadrants is a key part of working with angles on the coordinate plane. The coordinate plane is divided into four regions, or quadrants, by the x-axis and y-axis:

• The first quadrant is located where both x and y are positive.
• The second quadrant has negative x values and positive y values.
• In the third quadrant, both x and y are negative.
• The fourth quadrant features positive x values and negative y values.

Knowing the quadrant where an angle lies helps in determining its properties, such as the sign of its sine and cosine values. In the case of the exercise, the angle \( \frac{13\pi}{7} \) was found in the third quadrant, where both x and y coordinates are negative.

Radian Measure

Angles can be measured in degrees or radians. A radian is a measure based on the radius of a circle. One full circle is equivalent to \(2\pi\) radians, which is roughly 6.28 radians. This can be set side by side with degrees, where a full circle is 360 degrees.

The angle in the exercise, \( \frac{13\pi}{7} \), is given in radians. It's helpful to remember key conversions:

• \( \pi \) radians equals 180 degrees.
• \( \frac{\pi}{2} \) radians is equivalent to 90 degrees.
• \(2\pi\) radians correspond to 360 degrees.

Knowing these can aid in visualizing angles easily when working with both radian and degree measurements.

Angle Sketching

Sketching angles helps visualize their positions in different quadrants. Start by drawing the x-y plane with the positive x-axis as the initial side. When sketching, rotations are measured:

• Counterclockwise for positive angles.
• Clockwise for negative angles.

For the angle \( \frac{13\pi}{7} \) from the exercise, identify where it lands on the circle. Here, it passes \( \pi \) radians (180 degrees) but falls short of \(2\pi\). Hence, the angle terminates in the third quadrant. This sketch aids in calculating the reference angle and understanding angle properties better.

Radian Conversion

Conversion between radians and degrees is often necessary. To convert an angle from radians to degrees, utilize the relationship that \( \pi \) radians equal 180 degrees:

• Multiply by \( \frac{180}{\pi} \) to convert radians to degrees.
• Multiply by \( \frac{\pi}{180} \) to convert degrees to radians.

For the angle \( \frac{13\pi}{7} \), its degree measure can be calculated by:\[ \frac{13\pi}{7} \times \frac{180}{\pi} \approx 334.28 \text{ degrees} \]This conversion can make understanding and visualizing angles more intuitive, especially when paired with the quadrant and sketching concepts discussed earlier.