Problem 3 - Contemporary Precalculus

Question

Find the radian measure of the angle in standard position formed by rotating the terminal side by the given amount. \(1 / 18\) of a circle

Step-by-Step Solution

Answer

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

Step 1: Identify the proportion of a full circle rotation given

The problem states that the terminal side of an angle is rotated \(1/18\) of a circle.

Step 2: Recall the radian measure of a full circle rotation

We know that one full circle rotation (360 degrees) is equal to \(2\pi\) radians.

Step 3: Calculate the radian measure of the given proportion of circle rotation

Since we want to find the radian measure for \(1/18\) of a circle rotation, we can set up a proportion: $$\frac{\text{radian measure}}{2\pi} = \frac{1}{18}$$ Now, to find the radian measure, we can cross-multiply and solve for it: $$\text{radian measure} = 2\pi \cdot \frac{1}{18}$$

Step 4: Simplify the expression to find the radian measure

Simplify the expression by multiplying the numerator and denominator by \(2\pi\): $$\text{radian measure} = \frac{2\pi}{18}$$ Now, we can simplify the fraction: $$\text{radian measure} = \frac{\pi}{9}$$ Therefore, the radian measure of the angle in standard position formed by rotating the terminal side by \(1/18\) of a circle is \(\frac{\pi}{9}\) radians.

Key Concepts

Circle Rotation

When we talk about circle rotation, we are referring to the action of turning a point or a line around the center of a circle. There's an important connection between the number of circle rotations and measuring angles in radians. A full circle rotation means moving completely around the circle, which represents 360 degrees or equivalently, 2\(\pi\) radians. Partial rotations, like \(1/18\) of a turn, mean doing a smaller portion of that full rotation.

Think of a circle like a pie that you divide into pieces. When an angle is created by turning the terminal side (the ending side) of a ray from its initial position at the x-axis to some position on the circle, we say the circle has been rotated by a certain amount. The angle formed, expressed usually in radians or degrees, tells us how far the rotation has gone.

To relate rotation to radians, it's vital to understand that

• One full circle is divided into 2\(\pi\) radians.
• Any fraction of a circle represents a fraction of those 2\(\pi\) radians.

That ability to express fractions of a rotation in terms of radians is why the concept of radian measure is so handy.

Angle in Standard Position

An angle in standard position is defined when its vertex is at the origin of a coordinate plane and its initial side lies along the positive x-axis. This is a basic concept in trigonometry and geometry, utilized heavily in problems involving circle rotations and angular measurements.

In standard position:

• The vertex of the angle is always at the origin (0, 0).
• The initial side starts on the positive x-axis.
• The terminal side is where the angle ends after rotation.

Imagine standing at the origin, and the terminal side swings out from the x-axis. The rotation direction can be counterclockwise or clockwise. Counterclockwise is considered a positive angle, while clockwise rotation is negative.

Understanding this concept is vital because it standardizes the positioning of angles, making it easier to compare, measure, and calculate them, especially when converting between different units of measure, such as degrees and radians.

Radians to Degrees Conversion

Converting between radians and degrees is a common task in mathematics, especially when dealing with angles in standard position and circle rotations. A full circle is traditionally measured as 360 degrees but in radian measure, it equals 2\(\pi\) radians. Hence, there is a direct conversion formula:

To convert radians to degrees:- Use the conversion factor \(\frac{180}{\pi}\), since 180 degrees equal \(\pi\) radians.
To convert degrees to radians:- Use the conversion factor \(\frac{\pi}{180}\), reversing the relationship.

For instance, if you have an angle of \(\frac{\pi}{9}\) radians, converting it to degrees involves multiplying:
\[\text{Degrees} = \frac{\pi}{9} \times \frac{180}{\pi} = 20\]
Thus, \(\frac{\pi}{9}\) radians is equivalent to 20 degrees. This process relies on using the equivalence of 180 degrees being \(\pi\) radians, a foundational truth in trigonometry.
Mastering these conversions allows a seamless transition between radians and degrees, enabling a deeper understanding of various mathematical and scientific applications.