How to find the sinθ, cosθ, cscθ from tanθ=5/9?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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`

embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

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)` .

`costheta=(adj)/(hyp)=9/sqrt(106)`

`sintheta=(opp)/(hyp)=5/sqrt(106)`

`csctheta=(hyp)/(opp)=sqrt(106)/5`

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