Understanding When to Use + / - in Pythagorean Identity

[CrossFit415]

Homework Statement

How would I know if let's say

Sec^2 = 5 / 4
Sec = +/- $$\sqrt{5/4}$$
Sec = + / - $$\sqrt{5}$$ / 2

Now how would I know if whether to put a negative sign or a positive sign before the answer?

Because the textbook says Sec Theta < 0 therefor they added the negative sign, only.
What does this mean?

How is Sec < 0 if it equals $$\sqrt{5}$$ / 2 ? This is not less than 0 so shouldn't we use a positive symbol?

Thanks

Homework Equations

The Attempt at a Solution

 

[eumyang]

First off, please don't write a trig function without the angle. It's sec θ, not sec.

A trig ratio is positive or negative depending on the angle. Was there any mention of the angle itself, or the location of the point on the terminal side? sec θ = r/x, and if sec θ < 0 then x must be negative. Must be an angle in Quadrant II or III, then.

The title of your thread is "Pythagorean Identity," so I'm guessing you're making use of
1 + tan²θ = sec²θ. Were you told anything about tan²θ? It would have been helpful if you posted the entire problem and not just a piece of it.

 

[CrossFit415]

I apologize.

"Given that tan θ = 1/2 and sin θ <0, find the exact value of each of the remaining five trigonometric functions of θ."

yes, they used;
1 + (tan^2) θ = (sec^2) θ

 

[Lostinthought]

as tan is +ve and sin is -ve,the angle lies in the 3rd quadrant where sec is negtive,so u shld take the -ve value