Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined.


sec (7π/6)

Verified step by step guidance

1. Recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
2. Determine the reference angle for \( \theta = \frac{7\pi}{6} \).
3. Since \( \frac{7\pi}{6} \) is in the third quadrant, the reference angle is \( \theta - \pi = \frac{\pi}{6} \).
4. In the third quadrant, cosine is negative, so \( \cos\left(\frac{7\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) \).
5. Use the known value \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \) to find \( \sec\left(\frac{7\pi}{6}\right) = \frac{1}{-\frac{\sqrt{3}}{2}} \).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and secant, relate angles to ratios of sides in right triangles. The secant function, specifically, is defined as the reciprocal of the cosine function. Understanding these functions is essential for evaluating expressions involving angles, particularly in the context of the unit circle.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It provides a geometric representation of trigonometric functions, where the x-coordinate corresponds to the cosine of an angle and the y-coordinate corresponds to the sine. By using the unit circle, one can easily determine the values of trigonometric functions for various angles, including those expressed in radians.

Angle Measurement in Radians

Radians are a unit of angular measure where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius. The angle 7π/6 radians corresponds to 210 degrees, which is located in the third quadrant of the unit circle. Understanding how to convert between degrees and radians is crucial for evaluating trigonometric functions accurately.