Problem 2 - Contemporary Precalculus

Question

Find the radian measure of the angle in standard position formed by rotating the terminal side by the given amount. \(1 / 24\) of a circle

Step-by-Step Solution

Answer

Answer: The radian measure of the angle in standard position is \(\frac{1}{12}\pi\).

Step 1: Write down the given information and the formula to find the radian measure of the angle

Given: \(1/24\) of a circle. Formula: Portion of a circle × Total radian measure of a circle = Radian measure of angle.

Step 2: Identify the total radian measure of a circle

A full circle contains \(360^\circ\). In radians, a circle has \(2\pi\) radians.

Step 3: Calculate the radian measure of the angle

Now, we can substitute the given information into the formula. Radian measure of angle = \(\frac{1}{24}(2\pi)\)

Step 4: Simplify and find the radian measure of the angle

Simplify the expression and calculate the radian measure: Radian measure of angle = \(\frac{1}{12}\pi\) The radian measure of the angle in standard position is \(\frac{1}{12}\pi\).

Key Concepts

Angle Measurement

Understanding angle measurement is essential to solving problems involving rotations and geometry. Angles can be measured in degrees or radians.

• Degrees: A full circle encompasses 360 degrees. This is a common unit used in many everyday contexts.
• Radians: This is a more mathematical unit, often used in higher mathematics, where a full circle is equal to \(2\pi\) radians.

Choosing between degrees and radians often depends on the context and convenience for the calculations. Radians are particularly useful for calculus and trigonometric functions, providing a direct relationship with the radius of circles. To convert degrees to radians, use the formula: \[\text{Radians} = \frac{\pi}{180} \times \text{Degrees}\]Conversely, to convert radians to degrees, use:\[\text{Degrees} = \frac{180}{\pi} \times \text{Radians}\]These conversions help in understanding and applying different units of angle measurement interchangeably.

Circle Geometry

Circle geometry revolves around calculating dimensions and properties related to circles.

• The radius is the distance from the center of the circle to any point on its boundary.
• The diameter, which is twice the radius, is the length of a line passing through the center, touching two points on the boundary.
• The circumference is the total distance around the circle, calculated as \(2\pi r\), where \(r\) is the radius.

In this exercise, we consider a fraction of a circle, specifying \(\frac{1}{24}\) of a circle. Each fraction of a circle corresponds to a certain angle in both degrees and radians. Because a full circle in radians is \(2\pi\), a portion of \(\frac{1}{24}\) of a circle corresponds to this portion of \(2\pi\). This is how circle geometry directly connects with angle measurements in radians.

Radians

Radians are a fundamental concept in trigonometry and mathematics. They provide a natural way of measuring angles based on the radius of a circle. One radian is the angle formed when the arc length is equal to the radius of the circle, making it dimensionless and a versatile unit.

• A full circle equals \(2\pi\) radians.
• Half a circle, or a straight angle, is \(\pi\) radians.
• A right angle, which is half of a straight angle, is \(\frac{\pi}{2}\) radians.

Radians simplify many mathematical expressions and derivations, such as in calculus, where differentiation and integration become more straightforward. In this exercise, finding the radian measure for an angle that is \(\frac{1}{24}\) of a circle involves calculating \(\frac{1}{24} \times 2\pi\), leading to \(\frac{\pi}{12}\). Thus, understanding and computing in radians provides a more seamless approach in advanced mathematics and technological applications.