find all values of theta (in radian) such that tan (2/3 theta) = - sqrt3, in the interval [0, 2pi)

Question

 find all values of theta (in radian) such that tan (2/3 theta) = - sqrt3, in the interval [0, 2pi)

Answer 1

Greetings! Lets solve this shall we ?

So the equation we have is

tan(2θ/3) = - √3...............We must find all values of theta in radians such that this holds true. Then, we begin to solve by taking the inverse tangent to both sides such that

2θ/3 = tan^-1( -√3)..............Note that θ = tan^-1(-√3) = 2π/3 and 5π/3..............Then we solve for theta using both of these such that

2θ/3 = 2π/3

θ = 2π/3 (3/2)

θ = π..............This is one of the theta values

Now we solve for theta such that

2θ/3 = 5π/3

θ = 5π/3 (3/2)

θ = 5π/2

I hope this helped!