how do i solve 4sec(x)+8=0

Question

I know i have to isolate the variable, and when i do that i get sec(x)= -2 but i dont know where to go from there.

Answer 1

You are half way there!

You have sec(x) = -2 .

cos(x) = 1/sec(x)

So sec(x) = -2 the same place cos(x) = -1/2

cos(x) = -1/2 => cos^-1(x) = -1/2

cos(x) is negative in the second and third quadrant . So using a reference angle of 60 (since cos (1/2) = 60º)

In the second and third quadrant you get 180º- 60º = 120º and 180º+ 60º = 240º ( You can also use your unit circle to easily find these values!)

So cos^-1(-1/2 ) = sec^-1(2) = 120º and 240º or 2π/3 and 4π/3

(To convert from degrees to radians multiply by π/180)

Answer 2

Since sec(x) = -2 ; and sec(x) = 1/cos(x) we have

cos(x) = -1/2

In the standard interval [0, 2 pi) there are two solutions

x = 2 pi/3 and x = 4 pi/3

Answer 3

sec(x) = -2

Take the reciprocal of both sides to get

1/sec(x) = -1/2

From your identities you get that 1/sec(x) = cos(x) therefore,

cos(x) = -1/2

Use the inverse or arc trigonometry to isolate x

x = arc cos (-1/2)

If you use degrees x = 120

If you use radians x = 2(pi)/3