How to find the sinθ, cosθ, cscθ from tanθ=5/9?
It is given that `tan theta = 5/9`
`tan theta = sin theta/cos theta`
=> `(sin^2 theta)/(cos^2 theta) = 25/81`
=> `81* sin^2 theta = 25(1 - sin^2 theta)`
=> `106*sin^2 theta = 25`
=> `sin theta = sqrt(25/106) = 5/sqrt 106`
`cos theta = (5/sqrt 106)/(5/9) = 9/sqrt 106`
`csc theta = 1/sin theta = sqrt 106/5`
The value of `sin theta = 5/sqrt 106` , `cos theta = 9/sqrt 106` and `csc theta = sqrt 106/5`
Another way is to draw a right traingle with acute angle `theta` : the leg opposite `theta` has length 5 and the leg adjacent to `theta` has length 9.
Then by the pythagorean theorem the hypotenuse has length `sqrt(106)` .