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=(14)/(16)+1 \\ \sec ^2\theta=(14+16)/(16) \\ \sec ^2\theta=(30)/(16) \end{gathered}`

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

`\begin{gathered} \sec ^{}\theta=\sqrt{(30)/(16)} \\ \sec ^{}\theta=-\frac{\sqrt[]{30}}{4} \end{gathered}`