Question

which of the following could be trigonometric functions of the same angle?
○ cot y = 1/5, tan y = 5, cos y = 1/6
○ sin y = 3/5, tan y = 3/4, cot y = 4/3
○ csc y = 41/9, sin y = 9/41, cot y = 40/9
○ cos y = 8/17, cot y = 8/15, sec y = 17/15

Explanation:

Step1: Recall trigonometric identities

We know that $\tan Y=\frac{\sin Y}{\cos Y}$, $\cot Y = \frac{1}{\tan Y}$, $\csc Y=\frac{1}{\sin Y}$, $\sec Y=\frac{1}{\cos Y}$.

Step2: Check each option

For the first - option: If $\cos Y=\frac{8}{17}$, then $\sec Y=\frac{17}{8}$, $\cot Y=\frac{\cos Y}{\sin Y}$. Using the Pythagorean identity $\sin^{2}Y + \cos^{2}Y=1$, $\sin Y=\sqrt{1 - \cos^{2}Y}=\sqrt{1 - (\frac{8}{17})^{2}}=\frac{15}{17}$, and $\cot Y=\frac{8}{15}$.
For the second - option: If $\csc Y=\frac{41}{9}$, then $\sin Y=\frac{9}{41}$, and $\cot Y=\frac{\cos Y}{\sin Y}$. Using $\sin^{2}Y+\cos^{2}Y = 1$, $\cos Y=\sqrt{1-\sin^{2}Y}=\sqrt{1 - (\frac{9}{41})^{2}}=\frac{40}{41}$, and $\cot Y=\frac{40}{9}$.
For the third - option: If $\sin Y=\frac{3}{5}$, then $\tan Y=\frac{3}{4}$ (since $\cos Y=\frac{4}{5}$ using $\sin^{2}Y+\cos^{2}Y = 1$), and $\cot Y=\frac{4}{3}$.
For the fourth - option: If $\cot Y=\frac{1}{5}$, then $\tan Y = 5$. If $\tan Y = 5=\frac{\sin Y}{\cos Y}$, and $\sin^{2}Y+\cos^{2}Y = 1$. Let $\sin Y = 5\cos Y$, then $(5\cos Y)^{2}+\cos^{2}Y=1$, $25\cos^{2}Y+\cos^{2}Y=1$, $26\cos^{2}Y=1$, $\cos Y=\frac{1}{\sqrt{26}}$. This option has inconsistent values.

The first option is correct as when $\cos Y=\frac{8}{17}$, $\sin Y=\frac{15}{17}$, $\tan Y=\frac{15}{8}$, $\cot Y=\frac{8}{15}$, $\sec Y=\frac{17}{15}$.

Answer:

$\cos Y=\frac{8}{17}$, $\cot Y=\frac{8}{15}$, $\sec Y=\frac{17}{15}$