How do you solve tan2 theta equals 3 where the equation for theta is less than or equal to theta less than 2 pi?

Answer 1

Let us denote theta by the letter 't'

tan(2t)=3

2t=tan^-1(3)

2t=1.249 radian

t=1.249/2 = 0.6245 radian ----(1)

tan is positive in quadrant 3, so add PI to 1.249 to get another solution.

The other solution for t is 4.39

2t=4.39

t=4.39/2 = 2.195 radian ----(2)

The period of tan(2t) is PI/2. Therefore, the general solution is

0.6245+nPI/2, where n is any integer

That is , 0.6245+nPI/2, where n is any integer

Let n=1

t=2.19 -------------(3) (same as (2))

let n=2

t=3.766 -------------(4)

Let n=4

t=6.99 (exceeds 2PI) ---- 2PI= 2(3.14)=6.28

Now let us add nPI/2 to 2.195 radian

t=2.195+nPI/2

let n=1

t=3.766 -----------(5) (same as (4))

let n=2

t=5.3366 ---------(6)

The 4 different solutions are theta=0.6245, 2.195, 3.766, 5.3366