Problem 366 - Precalculus

Question

Find the angle between 0 and 2\(\pi\) in radians that is coterminal with \(-\frac{4 \pi}{7}\)

Step-by-Step Solution

Answer

The angle coterminal with \(-\frac{4\pi}{7}\) between 0 and 2\(\pi\) is \(\frac{10\pi}{7}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same terminal side. To find an angle coterminal with a given angle, you can add or subtract full rotations of the circle, which are given by multiples of \(2\pi\).

Step 2: Finding the Positive Coterminal Angle

To find an angle between 0 and \(2\pi\) that is coterminal with \(-\frac{4\pi}{7}\), add \(2\pi\) to \(-\frac{4\pi}{7}\). This addition will adjust the angle to be within the desired range:\[-\frac{4\pi}{7} + 2\pi = -\frac{4\pi}{7} + \frac{14\pi}{7} = \frac{10\pi}{7}\]

Step 3: Verifying the Range

Ensure that the angle \(\frac{10\pi}{7}\) is between 0 and \(2\pi\). Since \(2\pi = \frac{14\pi}{7}\), we check:\[0 < \frac{10\pi}{7} < \frac{14\pi}{7} \text{ which holds true.}\] This confirms the angle is indeed in the correct range.

Key Concepts

Radians

Radians are a way of measuring angles, often used in trigonometry and calculus. Unlike degrees, which divide one full rotation into 360 equal parts, radians use the radius of a circle to define an angle. In a circle with a radius of 1 unit, one full rotation around its circumference is equal to \(2\pi\) radians. This is because the circumference of a circle is calculated as \(2\pi\) times the radius. For context, \(\pi\) radians equals 180 degrees.

• A radian measures how far along the circular arc a given angle will take you.
• Radian measures are often more convenient when dealing with mathematical functions.
• Common angle conversions include \(\pi/2\) radians equating to 90 degrees.

Working with radians often simplifies mathematical calculations, particularly when dealing with trigonometric functions, calculus, and physics. When we are solving problems involving angles, using radians allows for straightforward operations like addition and subtraction, especially when finding coterminal angles.

Terminal Side

The terminal side of an angle in standard position is the side that moves or rotates from its initial position to form the angle. It is easy to visualize this if you think of the angle as being drawn in a circle with the center at the origin of a coordinate plane.
The initial side of the angle always starts along the positive x-axis.

• For a given angle, the terminal side could end up anywhere on the circle.
• Coterminal angles share the same terminal side. They differ by one or more full rotations around the circle.
• Because of this, any angle could have infinite coterminal angles, depending on whether you add or subtract multiples of \(2\pi\).

Understanding the concept of the terminal side is essential, especially when exploring how angles can be repeatedly rotated to find an equivalent angle, as we did by adding \(2\pi\) in the example exercise. This concept is central to trigonometry as it helps in determining angles that share the same values for the sine, cosine, and tangent functions.

Trigonometric Angles

Trigonometric angles are important when studying the functions sine, cosine, and tangent. These functions are foundational in the study of trigonometry and are deeply connected to the unit circle — a circle with radius 1 centered at the origin of a coordinate plane.

• The unit circle helps visualize and calculate trigonometric functions.
• An angle's position determines the values of the trigonometric functions for that angle.
• Angles that are coterminal (i.e., whose terminal sides coincide) have the same sine, cosine, and tangent values.

For example, angles \(-\frac{4\pi}{7}\) and \(\frac{10\pi}{7}\) are coterminal, and thus, they will yield the same results when used in trigonometric calculations. Mastering trigonometric angles offers insight into periodic phenomena like waves and oscillations and provides a powerful tool in many fields such as engineering and physics.