Question

Hi, can you help me to solve this exercise, please!

Answer 1

Trigonometric Identities.

To solve this problem, we need to keep in mind the following:

* The tangent function is negative in the quadrant II

* The cosine (and therefore the secant) function is negative in the quadrant II

* The tangent and the secant of any angle are related by the equation:

$\sec ^2\theta=\tan ^2\theta+1$

We are given:

$\text{tan}\theta=-\frac{\sqrt[]{14}}{4}$

And θ lies in the quadrant Ii.

Substituting in the identity:

$\begin{gathered} \sec ^2\theta=(-\frac{\sqrt[]{14}}{4})^2+1 \\ \text{Operating:} \\ \sec ^2\theta=\frac{14}{16}+1 \\ \sec ^2\theta=\frac{14+16}{16} \\ \sec ^2\theta=\frac{30}{16} \end{gathered}$

Taking the square root and writing the negative sign for the secant:

$\begin{gathered} \sec ^{}\theta=\sqrt{\frac{30}{16}} \\ \sec ^{}\theta=-\frac{\sqrt[]{30}}{4} \end{gathered}$