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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calendarEducator since 2011
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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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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)