Question

which of the following could be trigonometric functions of the same angle?
cos y = \frac{8}{17}, cot y = \frac{8}{15}, sec y = \frac{17}{15}
sin y = \frac{3}{5}, tan y = \frac{3}{4}, cot y = \frac{4}{3}
cot y = \frac{3}{5}, tan y = \frac{5}{3}, cos y = \frac{3}{8}
csc y = \frac{40}{9}, sin y = \frac{9}{41}, cot y = \frac{40}{9}

Explanation:

Step1: Recall the reciprocal relationship

Recall that \(\sec Y=\frac{1}{\cos Y}\). For the first option, if \(\cos Y = \frac{8}{17}\), then \(\sec Y=\frac{17}{8}
eq\frac{17}{15}\), so this option is wrong.

Step2: Use the reciprocal relationship for cotangent and tangent

Recall that \(\cot Y=\frac{1}{\tan Y}\). For the second option, if \(\tan Y=\frac{3}{4}\), then \(\cot Y = \frac{4}{3}\). Also, if \(\sin Y=\frac{3}{5}\), by \(\tan Y=\frac{\sin Y}{\cos Y}\), we can find \(\cos Y=\frac{4}{5}\), and these values are consistent with trigonometric identities.

Step3: Check the reciprocal relationship for cotangent and tangent (third option)

Since \(\cot Y=\frac{1}{\tan Y}\), if \(\cot Y=\frac{3}{5}\), then \(\tan Y=\frac{5}{3}
eq\frac{5}{3}\) (contradiction in the given values of \(\cot Y\) and \(\tan Y\) in the third - option statement as written: if \(\cot Y=\frac{3}{5}\), \(\tan Y\) should be \(\frac{5}{3}\), but the option has \(\tan Y = \frac{5}{3}\) which is wrong in the context of the given \(\cot Y=\frac{3}{5}\) as per the problem's numbering. Also, using \(\tan Y=\frac{\sin Y}{\cos Y}\) and Pythagorean identity \(\sin^{2}Y+\cos^{2}Y = 1\), if \(\tan Y=\frac{5}{3}\), let \(\sin Y = 5k\) and \(\cos Y=3k\), then \((5k)^{2}+(3k)^{2}=1\), \(25k^{2}+9k^{2}=1\), \(34k^{2}=1\), \(k^{2}=\frac{1}{34}\). And if \(\cos Y=\frac{3}{8}\), it will not satisfy the identity.

Step4: Check the reciprocal relationship for cosecant and sine

Recall that \(\csc Y=\frac{1}{\sin Y}\). For the fourth option, if \(\sin Y=\frac{9}{41}\), then \(\csc Y=\frac{41}{9}
eq\frac{40}{9}\), so this option is wrong.

Answer:

\(\sin Y=\frac{3}{5},\tan Y = \frac{3}{4},\cot Y=\frac{4}{3}\)