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
asked 2021-02-09
Given the following information about one trigonometric function, evaluate the other five functions.
\(\displaystyle{\cos{{u}}}=\frac{{5}}{{13}}\) , where \(\displaystyle{0}\le{u}\le\frac{\pi}{{2}}.\)

Answers (1)

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.
\(\displaystyle{\tan{{u}}}=\frac{{{\sin{{u}}}}}{{{\cos{{u}}}}}\)
\(\displaystyle=\frac{{\frac{{12}}{{13}}}}{{\frac{{5}}{{13}}}}\)
\(\displaystyle=\frac{{12}}{{5}}\)
\(\displaystyle{\sec{{u}}}=\frac{{1}}{{{\cos{{u}}}}}\)
\(\displaystyle=\frac{{1}}{{\frac{{5}}{{13}}}}\)
\(\displaystyle=\frac{{13}}{{5}}\)
\(\displaystyle{\cos{{e}}}{c}{u}=\frac{{1}}{{{\sin{{u}}}}}\)
\(\displaystyle=\frac{{1}}{{\frac{{12}}{{13}}}}\)
\(\displaystyle=\frac{{13}}{{12}}\)
\(\displaystyle{\cot{{u}}}=\frac{{1}}{{{\tan{{u}}}}}\)
\(\displaystyle=\frac{{1}}{{\frac{{12}}{{5}}}}\)
\(\displaystyle=\frac{{5}}{{12}}\)
0

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