Problem

Find the remaining five trigonometic functions of $\theta$
\[
\cot \theta=\frac{4}{3}, \sin \theta> 0
\]
Complete the following table.
\[
\begin{array}{ll}
\sin \theta=\square & \csc \theta=\square \\
\cos \theta=\square & \sec \theta=\square \\
\tan \theta=\square & \cot \theta=\frac{4}{3}
\end{array}
\]
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Answer

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Answer

Final Answer: \[\begin{array}{ll} \sin \theta=\boxed{0.6} & \csc \theta=\boxed{1.67} \\ \cos \theta=\boxed{0.8} & \sec \theta=\boxed{1.25} \\ \tan \theta=\boxed{0.75} & \cot \theta=\boxed{1.33} \end{array}\]

Steps

Step 1 :We know that \(\cot \theta=\frac{4}{3}\), which is the reciprocal of \(\tan \theta\). So, \(\tan \theta=\frac{3}{4}\).

Step 2 :We also know that \(\tan \theta=\frac{\sin \theta}{\cos \theta}\). So, we can use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\sin \theta\) and \(\cos \theta\).

Step 3 :Since \(\sin \theta>0\), we are in the first or second quadrant where both \(\sin \theta\) and \(\cos \theta\) are positive.

Step 4 :Finally, we can find \(\csc \theta\) and \(\sec \theta\) which are the reciprocals of \(\sin \theta\) and \(\cos \theta\) respectively.

Step 5 :Final Answer: \[\begin{array}{ll} \sin \theta=\boxed{0.6} & \csc \theta=\boxed{1.67} \\ \cos \theta=\boxed{0.8} & \sec \theta=\boxed{1.25} \\ \tan \theta=\boxed{0.75} & \cot \theta=\boxed{1.33} \end{array}\]

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