Step 1

Given:

\(\displaystyle \cos{\theta}=\frac{24}{{25}},{270}^{\circ}<\theta<{360}^{\circ}\)

So the angle lies in fourth quadrant. In the fourth quadrant cosine and secant are positive all others are negative.

Pythagorean identity is used for finding sine value

\(\displaystyle{{\sin}^{2}\theta}+{{\cos}^{2}\theta}={1}\)

\(\displaystyle \sin{\theta}=\pm\sqrt{{{1}-{{\cos}^{2}\theta}}}\)

\(\displaystyle=\pm\sqrt{{{1}-{\left(\frac{24}{{25}}\right)}^{2}}}\)

\(\displaystyle=\pm\sqrt{{\frac{{{25}^{2}-{24}^{2}}}{{25}^{2}}}}\)

\(\displaystyle=-\frac{7}{{25}}\)

The sine value is negative since angle is in third quadrant.

Step 2

The tan value can be found as

\(\displaystyle \tan{\theta}=\frac{{ \sin{\theta}}}{{ \cos{\theta}}}\)

\(\displaystyle=\frac{{-\frac{7}{{25}}}}{{\frac{24}{{25}}}}\)

\(\displaystyle=-\frac{7}{{24}}\)

Step 3

Secant is reciprocal of cosine.

\(\displaystyle \sec{\theta}=\frac{1}{{ \cos{\theta}}}\)

\(\displaystyle=\frac{1}{{\frac{24}{{25}}}}\)

\(\displaystyle=\frac{25}{{24}}\)

Step 4

Cosecant is the reciprocal of sine.

\(\displaystyle \csc{\theta}=\frac{1}{{ \sin{\theta}}}\)

\(\displaystyle=\frac{1}{{-\frac{7}{{25}}}}\)

\(\displaystyle=-\frac{25}{{7}}\)

Step 5

Cot is the reciprocal of tan

\(\displaystyle \cot{\theta}=\frac{1}{{ \tan{\theta}}}\)

\(\displaystyle=\frac{1}{{-\frac{7}{{24}}}}\)

\(\displaystyle=-\frac{24}{{7}}\)