Find the values of the remaining trigonometric functions a theta from the given information: a) sin theta = 5/13 , cos theta = -12/13 b) cos theta = -3/5 , tan theta is less than zero

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`sin theta = 5/13 `

`cos theta = -12/13 `

`==gt cos theta = (sin theta)/(cos theta) = (5/13)/(-12/13) = -5/12 `

`==gt sec theta = 1/(cos theta) = 1/(-12/13)= -13/12 `

`==gt csc theta = 1/(sin theta)= 1/(5/13)= 13/5 `

`==gt cot theta = (cos theta)/(sin theta)= (-12/13)/(5/13)= -12/5`

 

b) `cos theta = -3/5`

     `tan theta lt 0`

`==gt sec theta = 1/(costheta)= 1/(-3/5) = -5/3`

Given that `tan theta lt 0 `

`==gt (sintheta)/(costheta) lt 0 `

`==gt (sin theta)/ (-3/5) lt 0 `

`==gt sin theta gt 0`

==> We know that:

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

` ==gt sin^2 theta = 1- cos^2 theta = 1 - (9/25)= 16/25 `

`==gt sin theta = +- 4/5`

But `sintheta gt 0 `

`==gt sin theta = 4/5 `

`==gt tan theta = (sintheta)/(cos theta) = (4/5)/(-3/5)= -4/3 `

`==gt cot theta = -3/4 `

`==gt csc theta = 1/(sintheta)= 1/(4/5)= 5/4`

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