Given that:

`sectheta= sqrt 2,cottheta=-1` and `-pi<theta<pi`

find the exact value of the angle θ in radians. Justify your answer.

Expert Answers

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Given `sec theta=sqrt(2), cot theta=-1` ; find the exact value for `-pi<theta<pi` :

For both `sec theta=sqrt(2)` and `cot theta=-1` the reference angle is `pi/4` .

(`sec theta=sqrt(2) ==> cos theta =sqrt(2)/2 ==> theta=pi/4` and `cot theta=-1 ==> sin theta=-costheta` which only occurs at multiples of `pi/4` )

Since sec>0 the angle lies in the first or fourth quadrants. Since cot<0 the angle lies in the second or fourth quadrants.

Therefore the angle lies in the fourth quadrant.

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The angle is `theta=-pi/4`

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The angle `theta` cannot be comprised between `pi` and `pi` , hence, since `sec theta = 1/(cos theta) = sqrt2` and `cot theta = (cos theta)/(sin theta) = -1` , thus `cos theta` is positive and `sin theta` negative, theta is an angle in quadrant 4, so `(3pi)/2 < theta < 2pi.`

The problem provides the information that `sec theta = sqrt 2,` such that:

`{(sec theta = 1/(cos theta)), (sec theta = sqrt 2):} => 1/(cos theta) = sqrt 2 => cos theta = 1/sqrt2 => cos theta = sqrt2/2`

The problem also provides the information that `cot theta = -1` , such that:

`{(cot theta = (cos theta)/(sin theta)), (cot theta = -1):} => (cos theta)/(sin theta) = -1`

Since `cos theta = sqrt2/2` yields:

`(sqrt2/2)/(sin theta) = -1 => sin theta = -sqrt2/2`

Since `cos theta = sqrt2/2` and `sin theta = -sqrt2/2` and `theta` is in quadrant 4, yields that `theta` has the following value in radians such that:

`theta = 2pi - pi/4 => theta = (7pi)/4`

Hence, evaluating the value of theta, in radians, under the given conditions, yields `theta = (7pi)/4.`

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