Help solving trigonometric equation for phi

Hi,

I really need help solving for phi in the following equation:

115 = 42 tan^2 (45 + (phi/2))

If someone could show the working and explain the steps involved that would be greatly appreciated.

Thanks in advance!

115 = 42 tan^2(45+ phi/2)

(115/42) = tan^2 (45+phi/2)
+/- sqrt(115/42) = tan (45 +phi/2)
45 + phi/2 = arctan (+/- sqrt(115/42))+n(180), for n=0, 1, 2..
phi/2 = arctan (+/- sqrt(115/42)) + n(180) - 45
phi = 2(arctan (+/- sqrt(115/42)) + n(180) - 45)

Thank you kindly

Quote:

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Originally Posted by ebaines

115 = 42 tan^2(45+ phi/2)

(115/42) = tan^2 (45+phi/2)
+/- sqrt(115/42) = tan (45 +phi/2)
45 + phi/2 = arctan (+/- sqrt(115/42))+n(180), for n=0, 1, 2..
phi/2 = arctan (+/- sqrt(115/42)) + n(180) - 45
phi = 2(arctan (+/- sqrt(115/42)) + n(180) - 45)

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I have understood the question but i fail to understand as to why you have added n(180)-45
as far as my understanding goes we have the solution as phi = 2 [+ - {arctan sqrt (115/42) } - 45]
in case we are interested in getting all the values between 0 and 360 then we can do so after computing first two values.

I include the n(180) term to take into account the fact the arctan function returns a value between -90 and +90 degrees, but a valid answer may be 180 degrees different than this. In this particular problem it turns out to not be necessary, as the factor of 2 in front of the arctan term turns n(180) into n(360). But it's good practice to include the n(180) term so you don't leave out potential answers.