# If  `csc theta = sqrt(u^2+169)/u ` , find the expresion for `cot theta`  and `sec theta` .

lemjay | High School Teacher | (Level 2) Senior Educator

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`csc theta =sqrt(u^2+169)/u`

To determine the expression for cotangent and secant, draw the corresponding right triangle of the given cosecant function. To do so, apply its formula which is:

`csc theta= (hypoten u se)/(opposite)`

So, the hypotenuse of the right triangle is `sqrt(u^2+169)` and the side opposite to angle theta is u.

To determine the side adjacent to theta, apply the Pythagorean formula which is:

`c^2=a^2+b^2`

Plug-in c=sqrt(u^2+169)  and b=u.

`(sqrt(u^2+169))^2=a^2+u^2`

`u^2+169=a^2+u^2`

`u^2-u^2+169=a^2+u^2-u^2`

`169=a^2`

`sqrt169=sqrt(a^2)`

`13=a`

Hence, the adjacent side is 13.

Now that the three sides of the triangles are known (refer figure below), expression for cotangent and secant can now be determined.

For cotangent, apply its formula which is the ratio of adjacent side to the opposite side of theta.

`cot theta = 13/u`

And for secant, it is the ration of hypotenuse to adjacent side of angle theta.

`sec theta=sqrt(u^2+169)/13`

Hence:

`cot theta = 13/u `       and       `sec theta=sqrt(u^2+169)/13`

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