Problem 15 - Contemporary Precalculus

Question

Determine whether or not the given angles in standard position are coterminal. $$\frac{5 \pi}{12}, \frac{17 \pi}{12}$$

Step-by-Step Solution

Answer

Answer: No, the given angles are not coterminal.

Step 1: Write the difference between the angles

Write the difference between the given angles, which is: $$\frac{17\pi}{12} - \frac{5\pi}{12}$$

Step 2: Simplify the difference (subtract)

Minus the two fractions by finding the difference: $$\frac{12\pi}{12}$$

Step 3: Check if the difference is a multiple of 2π

Notice that $$\frac{12\pi}{12} = \pi$$, which is not a multiple of $$2\pi$$. To be coterminal, the fraction must be in the form $$2\pi k$$, where $$k$$ is an integer. Thus, the given angles are not coterminal. So, the given angles $$\frac{5\pi}{12} $$ and $$\frac{17\pi}{12}$$ are not coterminal.

Key Concepts

Understanding Radians

Radians are a unit of measure for angles, used primarily in trigonometry. Unlike degrees, which divide a circle into 360 parts, radians are based on the radius of a circle. This approach makes calculations involving circular arcs and sectors simpler.

• One full circle equals to \(2\pi\) radians.
• Half a circle is \(\pi\) radians.
• Radians allow us to express angles in terms of \(\pi\).

For the example provided, \(\frac{5\pi}{12}\) and \(\frac{17\pi}{12}\) are both measured in radians. These angles can be understood as fractions of \(\pi\) which relates directly to the circle's radius. Understanding radian measurement is fundamental for working with trigonometric concepts, as it streamlines various mathematical operations.

Angle Measurement in Standard Position

Angles measured in standard position begin with their initial side on the positive x-axis in a coordinate plane. The vertex is placed at the origin, and the terminal side is determined by the angle's measure. Standard position helps us easily assess and compare different angles' measures through straightforward use of coordinate systems.

• Positive angles turn counterclockwise from the initial side.
• Negative angles turn clockwise.

In the exercise case, both \(\frac{5\pi}{12}\) and \(\frac{17\pi}{12}\) can be visualized starting from the positive x-axis, determining the position of their terminal sides within the circle. Recognizing these positions and their comparisons is crucial in determining if they are coterminal, or reach the same endpoint on the coordinate system.

The Role of Trigonometry and Coterminal Angles

Trigonometry studies relationships in triangles, and by extension, circular functions and angles. In the context of this problem, coterminal angles are those which share the same terminal side when in standard position, yet may have different measurements.To determine if angles are coterminal:

• Calculate the difference between the angles.
• If the difference is a multiple of \(2\pi\), the angles are coterminal.

As calculated, the difference here \(\frac{17\pi}{12} - \frac{5\pi}{12} = \pi\) is not a multiple of \(2\pi\). Hence, these angles are not coterminal. Understanding these concepts helps in analyzing periodic behavior in functions and in broader mathematical applications.