Question

What is the exact value of secant of 3 times pi all over 4 question mark

Answer 1

The trigonometric identity secant is equal to:

`\sec \alpha=(1)/(\cos \alpha)`

In the unit circle, the cosine is given by the x-coordinate, then, here we have the unit circle:

As we said before, the cosine of an angle is the x-coordinate, then:

`\cos (3\pi)/(4)=\frac{-\sqrt[]{2}}{2}`

And, then the secant is:

`\begin{gathered} \sec (3\pi)/(4)=(1)/(\cos (3\pi)/(4)) \\ \sec (3\pi)/(4)=\frac{1}{\frac{-\sqrt[]{2}}{2}} \\ By\text{ applying the properties of fractions:} \\ \sec (3\pi)/(4)=\frac{2}{-\sqrt[]{2}} \\ \sec (3\pi)/(4)=\frac{2}{-\sqrt[]{2}}\cdot\frac{-\sqrt[]{2}}{-\sqrt[]{2}} \\ \sec (3\pi)/(4)=\frac{-2\sqrt[]{2}}{2} \\ \text{Simplify 2/2=1} \\ \sec (3\pi)/(4)=-\sqrt[]{2} \end{gathered}`

The answer is the last option.