Question

How do you find the solution to `tan^2theta+4tantheta-12=0` if `0<=theta<360`?

Answer 1

Solution: In `0 <=theta <=360 ,theta=63.43^0 , 99.46^0 ,243.43^0,279.46^0`

Explanation:

`tan^2 theta +4 tan theta -12 =0 or tan^2 theta +6 tan theta -2 tan theta -12 =0 or tan theta (tan theta + 6) -2 (tan theta +6) = 0` or
`(tan theta + 6) (tan theta -2) =0 :. (tan theta + 6)=0 or (tan theta -2) =0`
Case 1 : `tan theta + 6 =0 or tan theta = -6 :. theta = tan^-1(-6) =-80.54^0 =(360-80.54)=279.46^0`, also `theta =279.46-180= 99.46^0`[Since `tan` is negative in 2nd and 4th quadrant]

Case 2 : `tan theta - 2 =0 or tan theta = 2 :. theta = tan^-1( 2) =63.43^0`, also `theta =180+63.43= 243.43^0`[Since `tan` is positive in 1st and 3rd quadrant]

Solution: In `0 <=theta <=360 ,theta=63.43^0 , 99.46^0 ,243.43^0,279.46^0` [Ans]