find the remaining five trigonometric functions of \\( \\theta \\).\n\\ \n\\cot \\theta=\\frac{3}{4}, \\sin…

Explanation:

Step1: Determine the quadrant

Since (\cot\theta=\frac{3}{4}>0) and (\sin\theta < 0), (\theta) is in the third quadrant. In the third quadrant, (\sin\theta<0), (\cos\theta < 0), (\tan\theta>0), (\cot\theta>0), (\csc\theta<0), (\sec\theta<0).

Step2: Use the identity (\cot\theta=\frac{\cos\theta}{\sin\theta}=\frac{3}{4}), so (\cos\theta=\frac{3}{4}\sin\theta)

Also, use the Pythagorean identity (\sin^{2}\theta+\cos^{2}\theta = 1) Substitute (\cos\theta=\frac{3}{4}\sin\theta) into (\sin^{2}\theta+\cos^{2}\theta = 1) [ \begin{align*} \sin^{2}\theta+\left(\frac{3}{4}\sin\theta\right)^{2}&=1\ \sin^{2}\theta+\frac{9}{16}\sin^{2}\theta&=1\ \frac{16\sin^{2}\theta + 9\sin^{2}\theta}{16}&=1\ \frac{25\sin^{2}\theta}{16}&=1\ \sin^{2}\theta&=\frac{16}{25}\ \sin\theta&=-\frac{4}{5} \end{align*} ]

Step3: Find (\cos\theta)

Since (\cos\theta=\frac{3}{4}\sin\theta) and (\sin\theta =-\frac{4}{5}), then (\cos\theta=\frac{3}{4}\times\left(-\frac{4}{5}\right)=-\frac{3}{5})

Step4: Find (\tan\theta)

(\tan\theta=\frac{1}{\cot\theta}=\frac{4}{3})

Step5: Find (\csc\theta)

(\csc\theta=\frac{1}{\sin\theta}=-\frac{5}{4})

Step6: Find (\sec\theta)

(\sec\theta=\frac{1}{\cos\theta}=-\frac{5}{3})

Answer:

(\sin\theta=-\frac{4}{5}), (\cos\theta =-\frac{3}{5}), (\tan\theta=\frac{4}{3}), (\csc\theta=-\frac{5}{4}), (\sec\theta=-\frac{5}{3})