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:
[tex]\sec ^2\theta=\tan ^2\theta+1[/tex]
We are given:
[tex]\text{tan}\theta=-\frac{\sqrt[]{14}}{4}[/tex]
And θ lies in the quadrant Ii.
Substituting in the identity:
[tex]\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}[/tex]
Taking the square root and writing the negative sign for the secant:
[tex]\begin{gathered} \sec ^{}\theta=\sqrt{\frac{30}{16}} \\ \sec ^{}\theta=-\frac{\sqrt[]{30}}{4} \end{gathered}[/tex]
