[tex]\begin{gathered} \text{ Given} \\ \sin\theta=\frac{1}{4} \\ \cos\theta<0 \end{gathered}[/tex]
Since the sine function corresponds to the y values, and cosine function corresponds to x value, the quadrant for which x is negative, and y is positive is quadrant II. Knowing this information, we can have the following diagram.
The secant function is the inverse of the cosine function , which means we need to find the adjacent side. Using the Pythagorean Theorem, we have
[tex]\begin{gathered} a^2+b^2=c^2 \\ a^2+1^2=4^2 \\ a^2+1=16 \\ a^2=16-1 \\ a^2=15 \\ \sqrt{a^2}=\sqrt{15} \\ a=\sqrt{15} \end{gathered}[/tex]
This means that the adjacent side is √15.
Find now sec Θ
[tex]\begin{gathered} \sec\theta=\frac{\text{hypotenuse}}{\text{adjacent}} \\ \sec\theta=\frac{4}{\sqrt{15}} \\ \text{Rationalize the ratio and we get} \\ \sec\theta=\frac{4}{\sqrt{15}}\cdot\frac{\sqrt{15}}{\sqrt{15}} \\ \; \\ \text{Therefore,} \\ \sec\theta=\frac{4\sqrt{15}}{15} \end{gathered}[/tex]
