Question

find the exact value of \\( \sec ( - \pi ) \\). do not use a calculator.

\\( \sec ( - \pi ) = \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.

Explanation:

Step1: Use the property of secant function

The secant function is an even function, so \(\sec(-x)=\sec(x)\). Then \(\sec(-\pi)=\sec(\pi)\).

Step2: Use the reciprocal identity

Recall that \(\sec(x)=\frac{1}{\cos(x)}\). So \(\sec(\pi)=\frac{1}{\cos(\pi)}\).

Step3: Find the value of \(\cos(\pi)\)

We know that on the unit - circle, for the angle \(x = \pi\) (the point \((- 1,0)\) on the unit - circle), \(\cos(\pi)=-1\).

Step4: Calculate \(\sec(\pi)\)

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

Answer:

\(-1\)