Question

which of the following could be trigonometric functions of the same angle?
o csc y = 41/9, sin y = 40/41, cot y = 9/40
o cot y = 5/2, tan y = 2/5, cos y = 2/7
o sin y = 3/5, tan y = 4/3, cot y = 3/4
o cos y = 8/17, cot y = 8/15, sec y = 17/8

Explanation:

Step1: Recall trigonometric identities

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

Step2: Check the first - option

If $\cos Y=\frac{8}{17}$, then $\sec Y=\frac{17}{8}$. Using the Pythagorean identity $\sin^{2}Y = 1-\cos^{2}Y=1 - (\frac{8}{17})^{2}=1-\frac{64}{289}=\frac{289 - 64}{289}=\frac{225}{289}$, so $\sin Y=\frac{15}{17}$ (assuming $Y$ is in the first - quadrant where all trigonometric functions are positive for simplicity, since we are just checking identities). Then $\tan Y=\frac{\sin Y}{\cos Y}=\frac{15/17}{8/17}=\frac{15}{8}$ and $\cot Y=\frac{8}{15}$. This option has incorrect values for $\cot Y$ and $\sec Y$.

Step3: Check the second - option

If $\sin Y=\frac{3}{5}$, then $\csc Y=\frac{5}{3}$. Using the Pythagorean identity $\cos^{2}Y=1 - \sin^{2}Y=1 - (\frac{3}{5})^{2}=1-\frac{9}{25}=\frac{16}{25}$, so $\cos Y=\frac{4}{5}$. Then $\tan Y=\frac{\sin Y}{\cos Y}=\frac{3/5}{4/5}=\frac{3}{4}$ and $\cot Y=\frac{4}{3}$. This option has incorrect values for $\tan Y$ and $\cot Y$.

Step4: Check the third - option

If $\cot Y=\frac{2}{5}$, then $\tan Y=\frac{5}{2}$. Using the identity $1+\cot^{2}Y=\csc^{2}Y$ and $1 + \tan^{2}Y=\sec^{2}Y$. Also, $\tan Y=\frac{\sin Y}{\cos Y}=\frac{5}{2}$, so $\sin Y=\frac{5}{\sqrt{29}}$ and $\cos Y=\frac{2}{\sqrt{29}}$ (from $\sin^{2}Y+\cos^{2}Y = 1$ and $\frac{\sin Y}{\cos Y}=\frac{5}{2}$). This option is consistent with the trigonometric identities.

Step5: Check the fourth - option

If $\csc Y=\frac{41}{9}$, then $\sin Y=\frac{9}{41}$. Using the Pythagorean identity $\cos^{2}Y=1-\sin^{2}Y=1 - (\frac{9}{41})^{2}=1-\frac{81}{1681}=\frac{1681 - 81}{1681}=\frac{1600}{1681}$, so $\cos Y=\frac{40}{41}$. Then $\cot Y=\frac{\cos Y}{\sin Y}=\frac{40/41}{9/41}=\frac{40}{9}$. This option has incorrect values for $\cot Y$.

Answer:

$\cot Y=\frac{2}{5},\tan Y=\frac{5}{2},\cos Y=\frac{2}{\sqrt{29}}$ (the third - option values are consistent with trigonometric identities)