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


sec (5π/2)

Verified step by step guidance

1. Recognize that $\sec (\theta )=\frac{1}{\cos (\theta )}$. Therefore, we need to find $\cos \left(\frac{5\pi }{2}\right)$.
2. Convert $\frac{5\pi }{2}$ into degrees or a more familiar angle by noting that $2\pi$ is a full circle (360 degrees).
3. Subtract $2\pi$ from $\frac{5\pi }{2}$ to find the equivalent angle within the first circle: $\frac{5\pi }{2}-2\pi =\frac{\pi }{2}$.
4. Recognize that $\frac{\pi }{2}$ radians corresponds to 90 degrees, where the cosine value is 0.
5. Since $\cos \left(\frac{\pi }{2}\right)=0$, $\sec \left(\frac{5\pi }{2}\right)=\frac{1}{0}$, which is undefined.

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 crucial for evaluating expressions involving angles, especially 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 interpretation of trigonometric functions, where the x-coordinate represents the cosine and the y-coordinate represents the sine of an angle. Knowing how to locate angles on the unit circle helps in determining the values of trigonometric functions for various angles.

Angle Measurement and Periodicity

Angles can be measured in degrees or radians, with radians being the standard unit in calculus. The periodic nature of trigonometric functions means they repeat their values at regular intervals. For example, the secant function has a period of 2π, which is essential for evaluating angles like 5π/2, as it can be simplified to an equivalent angle within the standard range.