Question

Assume that `theta` is an acute angle in a right triangle satisfying the given condition. Evaluate the remaining trigonometric functions ? `sin theta= 4/11`

Find
`cos theta=`
`tan theta=`
`csc theta=`
`sec theta=`
`cot theta=`

Answer 1

See explanation.

Explanation:

First we can notice that all the values are positive because the angle is accute (i.e. it is located in `Q1`).

To calculate `costheta` we can use the identity saying that for any angle we have:

`sin^2theta+cos^2theta=1`

If we use the given value we get:

`(4/11)^2+cos^2theta=1`

`cos^2theta=1-(4/11)^2`

`cos^2theta=1-16/121=105/121`

`costheta=sqrt(105)/11`

Now having `sintheta` and `costheta` we can calculate the remaining functions:

`tantheta=sintheta/costheta` ##

`tantheta=4/11-:sqrt(105)/11=4/11*11/sqrt(105)=4/sqrt(105)=(4sqrt(105))/105`

`cottheta=1/tantheta=sqrt(105)/4`

`sectheta=1/costheta=11/sqrt(105)=(11sqrt(105))/105`

`csctheta=1/sintheta=11/4=2 3/4`