Questions: sec (-t+2 π)

Transcript text: $\sec (-t+2 \pi)$

Solution

Solution Steps

To solve the expression $\sec(-t + 2\pi)$, we can use the periodic properties of the trigonometric functions. The secant function, $\sec(x)$, is the reciprocal of the cosine function, $\cos(x)$. The cosine function is periodic with a period of $2\pi$, meaning $\cos(x) = \cos(x + 2\pi k)$ for any integer $k$. Therefore, $\sec(-t + 2\pi) = \sec(-t)$.

Solution Approach

Recognize that $\sec(x) = \frac{1}{\cos(x)}$.
Use the periodic property of cosine: $\cos(-t + 2\pi) = \cos(-t)$.
Therefore, $\sec(-t + 2\pi) = \sec(-t)$.

Step 1: Recognize the Periodicity of the Cosine Function

The cosine function is periodic with a period of $2\pi$. This means that for any angle $x$, $\cos(x) = \cos(x + 2\pi k)$ for any integer $k$. Therefore, $\cos(-t + 2\pi) = \cos(-t)$.

Step 2: Express Secant in Terms of Cosine

The secant function is defined as the reciprocal of the cosine function: \[ \sec(x) = \frac{1}{\cos(x)} \] Thus, $\sec(-t + 2\pi) = \frac{1}{\cos(-t + 2\pi)} = \frac{1}{\cos(-t)} = \sec(-t)$.

Step 3: Simplify Using Even Function Property

The cosine function is an even function, meaning $\cos(-x) = \cos(x)$. Therefore, $\sec(-t) = \frac{1}{\cos(-t)} = \frac{1}{\cos(t)} = \sec(t)$.