Problem 32 - Algebra 2

Question

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(\frac{7 \pi}{6}\)

Step-by-Step Solution

Answer

Positive: \(\frac{19\pi}{6}\), Negative: \(-\frac{5\pi}{6}\).

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same initial side and terminal side, but differ in the number of complete rotations. They can be found by adding or subtracting complete rotations (in radians, multiples of \(2\pi\)) from the given angle.

Step 2: Find the Positive Coterminal Angle

To find a positive coterminal angle for \(\frac{7\pi}{6}\), add \(2\pi\) to the original angle:\[\frac{7\pi}{6} + 2\pi = \frac{7\pi}{6} + \frac{12\pi}{6} = \frac{19\pi}{6}\]So, \(\frac{19\pi}{6}\) is a positive coterminal angle to \(\frac{7\pi}{6}\).

Step 3: Find the Negative Coterminal Angle

To find a negative coterminal angle for \(\frac{7\pi}{6}\), subtract \(2\pi\) from the original angle:\[\frac{7\pi}{6} - 2\pi = \frac{7\pi}{6} - \frac{12\pi}{6} = -\frac{5\pi}{6}\]Thus, \(-\frac{5\pi}{6}\) is a negative coterminal angle to \(\frac{7\pi}{6}\).

Key Concepts

Angle Measurement

Angles are a fundamental aspect of geometry, and measuring them accurately is crucial in various fields such as mathematics and physics.
An angle is typically measured in degrees or radians. While degrees are more commonly used in everyday situations, radians are the standard unit in mathematics.
To measure an angle, we consider two rays with a common endpoint, called the vertex. The amount of rotation from one ray to another determines the angle's measurement. Angles can range from 0° to 360° or from 0 to 2π radians.

• A full circle equals 360°.
• A full rotation also equals 2π radians.
• Coterminal angles occur when differing by full rotations of 360° (or 2π radians).

Understanding angle measurement in radians is essential for understanding coterminal angles, as it allows us to find equivalent angles by adding or subtracting multiples of 2π radians.

Radian

The radian measures angles based on the arc length they subtend on a unit circle. One radian is the angle subtended by an arc that is equal in length to the circle's radius.
This measurement helps maintain a direct relationship between linear and angular quantities.Radians are fundamental in calculus and trigonometry due to their natural integration with functions.
Converting between degrees and radians is straightforward:

• 1 radian ≈ 57.2958 degrees.
• 1 degree ≈ 0.01745 radians.
• To convert degrees to radians, multiply by π/180.
• To convert radians to degrees, multiply by 180/π.

Using radians, we can easily determine coterminal angles by adding or subtracting \(2\pi\). If you have an angle of \(\frac{7\pi}{6}\), adding \(2\pi\) gives a positive coterminal angle, while subtracting \(2\pi\) results in a negative one.

Positive and Negative Angles

In trigonometry, angles can be defined as positive or negative based on their direction of rotation.Positive angles are formed by counterclockwise rotation, while negative angles result from clockwise rotation.
This directional distinction allows us to represent multiple angles that share the same terminal position on a circle, known as coterminal angles.Coterminal angles are crucial in simplifying complex trigonometric problems and can be found by adjusting the given angle by multiples of a full rotation \((2\pi)\).

• Positive coterminal angle: add \(2\pi\) to the given angle.
• Negative coterminal angle: subtract \(2\pi\).

For example, from \(\frac{7\pi}{6}\), \(\frac{19\pi}{6}\) is a positive coterminal angle and \(-\frac{5\pi}{6}\) is a negative coterminal angle. This understanding is key in various applications of trigonometric identities and equations.