Problem 1 - Contemporary Precalculus

Question

Find the radian measure of the angle in standard position formed by rotating the terminal side by the given amount. \(1 / 9\) of a circle

Step-by-Step Solution

Answer

Answer: The radian measure of the angle is \(\frac{2\pi}{9}\).

Step 1: Find the total radians in a circle

A full circle has a total of \(2\pi\) radians.

Step 2: Calculate 1/9th of the total radians

To find the radian measure of the angle when the terminal side is rotated by 1/9 of a circle, we need to multiply the total radians by 1/9. So the calculation is: \(\text{Radian Measure} = \frac{1}{9} \times 2\pi\)

Step 3: Simplify the expression

Now, we just need to simplify the expression to get the radian measure: \(\text{Radian Measure} = \frac{2\pi}{9}\) So, the radian measure of the angle when the terminal side is rotated by 1/9 of a circle is \(\frac{2\pi}{9}\).

Key Concepts

Angle in Standard Position

Understanding angles in standard position is crucial in trigonometry. An angle in standard position is defined as being oriented in a specific way: the vertex of the angle is placed at the origin of a coordinate system, usually with the initial side along the positive x-axis. The rotation to form the terminal side defines the angle. This orientation helps in establishing a uniform method to describe angles. Angles in standard position can be positive or negative depending on the direction of the rotation.

• Positive Angle: Formed by rotating the initial side counterclockwise to the terminal side.
• Negative Angle: Formed by rotating the initial side clockwise to the terminal side.

By placing angles in a standard position, calculations related to rotation, such as determining angular measures in radians or degrees, become more straightforward and systematic.

Circle Rotation

Circle rotation plays a fundamental role in understanding angles and their measurements. When an angle is formed in a circle, it can be described by how much the terminal side has rotated from the initial side. An entire circle represents a full rotation, which is equivalent to one complete revolution.

• Full Rotation: Equal to 360 degrees or \(2\pi\) radians.
• Partial Rotation: Any fraction of a full rotation, often described in terms of degrees or radians.

For example, if we take \(\frac{1}{9}\) of a circle as in the exercise, this partial rotation can easily be translated into radians by finding \(\frac{1}{9}\) of \(2\pi\) radians, resulting in \(\frac{2\pi}{9}\) radians. Recognizing these portions helps in calculating exact measures for angles formed through specific rotations.

Trigonometry

Trigonometry is an essential branch of mathematics that explores the relationships between angles and the sides of triangles. One of the core concepts in trigonometry is the use of radian measure, which provides a natural way to describe angles constructed from rotations.

• Radians: A measure based on the radius of the circle, commonly used in higher mathematics because it relates angles directly to the properties of circles.
• Functions: Trigonometry heavily relies on functions like sine, cosine, and tangent, which describe the relationships between the angles and sides of a right triangle.

These concepts apply to various fields such as physics, engineering, and computer science, where knowing how far a point has rotated in radians gives precise information crucial for calculations. Understanding these trigonometric principles allows for analyzing and solving problems across a wide spectrum of mathematical contexts.