Let theta be an angle in quadrant IV such that sin theta = -5/13 Find exact values of SEC theta and TAN theta?
You need to remember that the value of tangent function in quadrant 4 is negative, such that:
`tan theta = sin theta/cos theta`
The problem provides the value of `sin theta` , hence you may find the value of `cos theta` using Pythagorean trigonometric identity such that:
`cos theta = +-sqrt(1 - sin^2 theta)`
Since the value of `cos theta` is positive in quadrant 4, you need to keep only positive value `cos theta = +sqrt(1 - (-5/13)^2).`
`cos theta = sqrt(1 - 25/169)`
`cos theta = sqrt ((169-25)/169)`
`cos theta = sqrt (144/169)`
`cos theta = 12/13`
Hence, you should substitute`-5/13` for `sin theta ` and `12/13` for `cos theta` to evaluate tan theta such that:
`tan theta = (-5/13)/(12/13) =gt tan theta = -5/12`
You need to remember that `sec theta = 1/cos theta` , hence you need to substitute 12/13 for `cos theta` such that:
`sec theta = 1/(12/13) =gt sec theta = 13/12`
Hence, evaluating `tan theta` and `sec theta ` in quadrant 4 yields `tan theta = -5/12 ` and `sec theta = 13/12` .