# Given the following information about one trigonometric function, evaluate the other five functions. cos u=5/13 , where 0 <= u <= pi/2.

Question
Trigonometric Functions
Given the following information about one trigonometric function, evaluate the other five functions.
$$\displaystyle{\cos{{u}}}=\frac{{5}}{{13}}$$ , where $$\displaystyle{0}\le{u}\le\frac{\pi}{{2}}.$$

2021-02-10
Angle is in first quadrant and we know that in first quadrant all six trigonometric functions are positive.
Given, $$\displaystyle{\cos{{u}}}=\frac{{5}}{{13}}$$
We have identity $$\displaystyle{{\sin}^{{2}}{x}}+{{\cos}^{{2}}{x}}={1}$$
Therefore,
$$\displaystyle{\sin{{u}}}=\sqrt{{{1}-{{\cos}^{{2}}{u}}}}$$
$$\displaystyle=\sqrt{{{1}-{\left(\frac{{5}}{{13}}\right)}^{{2}}}}$$
$$\displaystyle=\sqrt{{{1}-\frac{{25}}{{169}}}}$$
$$\displaystyle=\sqrt{{\frac{{{169}-{25}}}{{169}}}}$$
$$\displaystyle=\sqrt{{\frac{{144}}{{169}}}}$$
$$\displaystyle=\frac{{12}}{{13}}$$
Now, finding other trigonometric function.
$$\displaystyle{\tan{{u}}}=\frac{{{\sin{{u}}}}}{{{\cos{{u}}}}}$$
$$\displaystyle=\frac{{\frac{{12}}{{13}}}}{{\frac{{5}}{{13}}}}$$
$$\displaystyle=\frac{{12}}{{5}}$$
$$\displaystyle{\sec{{u}}}=\frac{{1}}{{{\cos{{u}}}}}$$
$$\displaystyle=\frac{{1}}{{\frac{{5}}{{13}}}}$$
$$\displaystyle=\frac{{13}}{{5}}$$
$$\displaystyle{\cos{{e}}}{c}{u}=\frac{{1}}{{{\sin{{u}}}}}$$
$$\displaystyle=\frac{{1}}{{\frac{{12}}{{13}}}}$$
$$\displaystyle=\frac{{13}}{{12}}$$
$$\displaystyle{\cot{{u}}}=\frac{{1}}{{{\tan{{u}}}}}$$
$$\displaystyle=\frac{{1}}{{\frac{{12}}{{5}}}}$$
$$\displaystyle=\frac{{5}}{{12}}$$

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