Question

the point given below is on the terminal side of an angle \\( \theta \\) in standard position. find the exact value of each of the six trigonometric functions of \\( \theta \\)
\\ (12,-9) \\
\\ \sin \theta=\frac{3}{5} \\
(simplify your answer type an exact answer, using radicals as needed use integers or fractions for any numbers in the expression )
\\ \cos \theta=\frac{4}{5} \\
(simplify your answer type an exact answer, using radicals as needed use integers or fractions for any numbers in the expression )
\\ \tan \theta=-\frac{3}{4} \\
(simplify your answer type an exact answer, using radicals as needed use integers or fractions for any numbers in the expression )
\\ \cot \theta=\square \\
(simplify your answer type an exact answer, using radicals as needed use integers or fractions for any numbers in the expression )

Explanation:

Step1: Recall the definition of cotangent

Cotangent of an angle \(\theta\) in a right - triangle (or using the coordinates \((x,y)\) of a point on the terminal side of the angle \(\theta\) with \(r=\sqrt{x^{2}+y^{2}}\)) is given by \(\cot\theta=\frac{\cos\theta}{\sin\theta}=\frac{x}{y}\) (where \(x = 12\) and \(y=-9\) for the point \((x,y)=(12,-9)\)).

Step2: Substitute the values of \(x\) and \(y\)

We know that \(\cot\theta=\frac{x}{y}\). Substituting \(x = 12\) and \(y=-9\), we get \(\cot\theta=\frac{12}{-9}\).

Step3: Simplify the fraction

Simplify \(\frac{12}{-9}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 3. \(\frac{12\div3}{-9\div3}=-\frac{4}{3}\).

Answer:

\(-\frac{4}{3}\)