Question

find the exact value of sec (-2π). do not use a calculator. sec (-2π)= (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression )

Explanation:

Step1: Recall secant - cosine relationship

$\sec(x)=\frac{1}{\cos(x)}$, so $\sec(- 2\pi)=\frac{1}{\cos(-2\pi)}$.

Step2: Use cosine's even - function property

Since $\cos(-\theta)=\cos(\theta)$, then $\cos(-2\pi)=\cos(2\pi)$.

Step3: Evaluate cosine of $2\pi$

The cosine function has a period of $2\pi$, and $\cos(2\pi) = 1$.

Step4: Calculate secant value

Substitute $\cos(-2\pi)=1$ into $\sec(-2\pi)=\frac{1}{\cos(-2\pi)}$, we get $\sec(-2\pi)=\frac{1}{1}=1$.

Answer:

$1$