Solve sec^2 theta= sec theta+2 using exact values over the following domains

Question

=-2pi<=x<2pi

=the real numbers

Answer 1

sec^2 theta= sec theta+2

 

1/cos^2(theta) = 1/cos(theta) + 2

 

Multiply both sides by cos^2(theta).

 

1 = cos(theta)  + 2cos^2(theta)

 

Let x = cos(theta)

 

2x^2 + x - 1 = 0

 

x = (1/4){-1 ± √[1-4(2)(-1)]} = (1/4)(-1 ± 3) = 1/2 or -1

 

For x = -1, cos(theta) = -1 so theta = ±π; add ±2πk (k=0,1,2,3,...) to extend over all real numbers

For x = 1/2, cos(theta) = 1/2 so theta = ±π/3; add ±2πk (k=0,1,2,3,...) to extend over all real numbers