Use the fact that the trigonometric functions are periodic to find the exact value of the expression. Do not use a calculator.
-$\sec \frac { 19 \pi } { 4 }$
A) $- \sqrt { 2 }$
B) $- \frac { 2 \sqrt { 3 } } { 3 }$
C) $\frac { \sqrt { 2 } } { 2 }$
D) $- 2$

Question

Quizplus · Accepted Answer

The secant function has a period of $2\pi$, so $\sec \frac{19\pi}{4} = \sec (\frac{19\pi}{4} - 4\pi) = \sec (\frac{19\pi}{4} - \frac{16\pi}{4}) = \sec (\frac{3\pi}{4})$. The secant is the reciprocal of the cosine, and $\cos (\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2}$, so $\sec (\frac{3\pi}{4}) = -\sqrt{2}$.

Use the Fact That the Trigonometric Functions Are Periodic to Find