In Exercises 35–60, find the reference angle for each angle.17𝜋6

Verified step by step guidance

1. First, understand that a reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis.
2. Convert the given angle from radians to degrees if necessary, but in this case, we will work directly with radians.
3. The given angle is \( \frac{17\pi}{6} \). Since \( \pi \) radians is equivalent to 180 degrees, \( 2\pi \) radians is a full circle (360 degrees).
4. Determine how many full circles (\( 2\pi \)) fit into \( \frac{17\pi}{6} \) by dividing \( \frac{17\pi}{6} \) by \( 2\pi \). This helps to find the equivalent angle within the first circle (0 to \( 2\pi \)).
5. Subtract the full circles from \( \frac{17\pi}{6} \) to find the equivalent angle within the first circle, then find the reference angle by determining the acute angle it makes with the x-axis.

Key Concepts

Reference Angle

The reference angle is the acute angle formed by the terminal side of a given angle and the x-axis. It is always measured as a positive angle and is typically between 0 and π/2 radians (0° and 90°). For angles greater than 360° or 2π radians, the reference angle helps simplify trigonometric calculations by relating them to angles within the first quadrant.

Angle Measurement

Angles can be measured in degrees or radians, with radians being the standard unit in trigonometry. One full rotation (360°) is equivalent to 2π radians. Understanding how to convert between these two units is essential for finding reference angles, especially when dealing with angles expressed in radians, such as 17π/6.

Quadrants of the Unit Circle

The unit circle is divided into four quadrants, each corresponding to specific ranges of angle measures. The first quadrant contains angles from 0 to π/2, the second from π/2 to π, the third from π to 3π/2, and the fourth from 3π/2 to 2π. Knowing which quadrant an angle lies in helps determine its reference angle and the sign of its trigonometric functions.