Solve for the exact solutions in the interval [ 0 , 2 π ) [ 0 , 2 π ) . If the equation has no solutions, respond with DNE.

Question

1/2tan²(x) =1- sec(x)

Answer 1

(1/2)tan²(x) =1- sec(x)

 

First, we need the equation to be in the same trig function.

From the Pythagorean Identities, we know tan²(x) + 1 = sec²(x).

Solving for tan²(x), we know tan²(x) = sec²(x) - 1.

Substitute this into our equation:

 

(1/2)[sec²(x) - 1] = 1- sec(x)     (multiply both sides by 2)

 

sec²(x) - 1 = 2 - 2sec(x)     (subtract 2 from and add 2sec(x) to both sides)

 

sec²(x) + 2sec(x) - 3 = 0    (factor into two binomials)

 

(sec(x) + 3)(sec(x) - 1) = 0  (apply Zero Product Property)

 

sec(x) + 3 = 0    

and      

sec(x) - 1 = 0 

 

sex(x) = -3        

and      

sec(x) = 1     (take reciprocal of both sides)

 

cos(x) = -1/3    

and    

cos(x) = 1

 

x = cos^-1(-1/3)  (make sure your calculator is in radians)

and

x = cos^-1(1)

 

Because cosine is negative in the 2nd and 3rd quadrants, we have two solutions to the first equation:

x = 1.9106 + 2πk (2nd quadrant with reference angle π-1.9106 = 1.2310)

x = (π + 1.2310) + 2πk   (3rd quadrant)

and

x = 0  + 2πk     where k is an integer  

 

over the interval [0, 2π) there are 3 solutions:

 

x ∈ {0, 1.9106, 4.3726}

 

Hope this helps!