Let theta be an angle in quadrant IV such that sin theta = -5/13   Find exact values of SEC theta and TAN theta?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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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` .

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