Problem 20 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Sketch the angle. Then find its reference angle. \(\frac{8 \pi}{3}\)

Step-by-Step Solution

Answer

The reference angle for \(\frac{8 \pi}{3}\) is \(\frac{\pi}{3}\)

Step 1: Plot the angle

First, recognize that each full rotation on a plane is \(2\pi\) radians. The given angle is \(\frac{8 \pi}{3}\), which is greater than \(2\pi\) (or \(\frac{6 \pi}{3}\)). This means we have completed one full rotation and an additional \(\frac{8 \pi}{3} - 2\pi = \frac{2 \pi}{3}\) which falls into the second quadrant.

Step 2: Find the reference angle

The reference angle is found by subtracting the given angle from \(\pi\) if the angle is in the second quadrant, \(2\pi\) if it is in the third, or subtracting the given angle from \(2\pi\) if it is in the fourth. Since our angle is in the second quadrant, it will be \(\pi - \frac{2 \pi}{3} = \frac{\pi}{3}\).

Key Concepts

Radians

Understanding radians is crucial to mastering trigonometry and sketching angles. A radian is a unit of angular measure used in mathematics and is the standard unit of angular measure used in many areas of mathematics. One radian is the angle created when the radius of a circle is wrapped along the circumference of the circle. In terms of degrees, one radian is roughly equal to 57.3 degrees.

A full circle has an angle of \(2\pi\) radians, which is equivalent to 360 degrees. This means that \(\pi\) radians equal 180 degrees, half of a full rotation. For example, an angle measured as \(\frac{8\pi}{3}\) radians is an indication that the angle has rotated beyond a full circle (\(2\pi\)) and we must calculate how much it exceeds that complete rotation. Rationalizing the relationship between radians and degrees will help you not only to interpret angles in trigonometry but also to convert between these units when necessary.

Sketching Angles

Sketching angles accurately is a practical skill in trigonometry that helps visualize problems and solutions. To sketch an angle measured in radians, such as \(\frac{8\pi}{3}\), you begin by understanding the full rotation of a circle in radians (\(2\pi\)). When the angle exceeds \(2\pi\), as in our example, it signifies that the angle has completed a full circle and has some extra rotation.

The step-by-step approach requires us to sketch a complete rotation (\(2\pi\)) and then add the remaining \(\frac{2\pi}{3}\) radians. Since \(\frac{2\pi}{3}\) radians is two-thirds of \(\pi\) radians, it takes us into the second quadrant. This method of breaking down the angle can really simplify the process of sketching more complex angles and can help when working with trigonometric functions that rely on the precise placement of angles within the coordinate system.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the side lengths and angles of triangles. It is a key component when working with reference angles, as seen in our example with the angle \(\frac{8\pi}{3}\). A reference angle is defined as the acute angle formed by the terminal side of the given angle and the x-axis.

To find a reference angle for any angle in radians, first identify the quadrant where the angle is located. Once the quadrant is determined, use basic subtraction to find the acute angle. In the case of \(\frac{8\pi}{3}\), which resides in the second quadrant, the reference angle is found by subtracting the angle from \(\pi\) radians: \(\pi - \frac{2\pi}{3} = \frac{\pi}{3}\). This reference angle is vital in simplifying complex trigonometric calculations and is used extensively in trigonometric identities and equations.