Find each exact function value. See Example 3.
sec π/6

Verified step by step guidance

1. Recall the definition of the secant function: \(\sec \theta = \frac{1}{\cos \theta}\).
2. Identify the angle given: \(\theta = \frac{\pi}{6}\) radians.
3. Find the cosine of \(\frac{\pi}{6}\). From the unit circle or special triangles, \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\).
4. Use the secant definition to write \(\sec \frac{\pi}{6} = \frac{1}{\cos \frac{\pi}{6}}\).
5. Substitute the cosine value into the expression: \(\sec \frac{\pi}{6} = \frac{1}{\frac{\sqrt{3}}{2}}\) and simplify the fraction.

Key Concepts

Understanding the Secant Function

The secant function, sec(θ), is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). To find sec(π/6), you first find cos(π/6) and then take its reciprocal. This relationship is fundamental for evaluating secant values exactly.

Exact Values of Trigonometric Functions at Special Angles

Certain angles like π/6, π/4, and π/3 have well-known exact trigonometric values derived from special right triangles. For π/6, cos(π/6) equals √3/2. Knowing these exact values allows precise calculation without approximations.

Using the Unit Circle for Angle Measurement

The unit circle represents angles in radians and their corresponding trigonometric values on a circle of radius 1. π/6 radians corresponds to 30 degrees, and locating this angle on the unit circle helps visualize and confirm the cosine and secant values.