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What values for theta (`0lt=thetalt=2pi` ) satisfy the equation. `costheta-tanthetacostheta=0` A)3pi/4, 7pi/4 b) 0, 5pi/4  c) 7pi/6, 11pi/6 D) pi/4, 5pi/4

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Luca B. eNotes educator | Certified Educator

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You also may approach the equation using factorization, such that:

`cos theta*(1 - tan theta) = 0`

Using zero product rule yields:

`{(cos theta = 0),(1 - tan theta = 0):} `

Since none of the answer provided does not include a value for theta that satisfies the equation `cos theta = 0` , hence, only `1 - tan theta = 0` , such that:

`1 - tan theta = 0 => -tan theta = -1 => tan theta = 1`

Since the tangent function is positive in quadrants 1 and 3 yields that `theta` can have the following values:

`theta = pi/4` (quadrant 1)

`theta = pi + pi/4 = (5pi/4)` (quadrant 3)

Hence, evaluating the correct answer among the options provided yields that option `D) pi/4, (5pi/4)` is valid.

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crmhaske eNotes educator | Certified Educator

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First simplify the problem:

`tantheta=sintheta/costheta`

`costheta-(sintheta/costheta)costheta=0`

`costheta-sintheta=0`

 

`costheta=sintheta `

The values for which cosine equals sine are pi/4 and 5pi/4; threfore, the answer is d)

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