Question

Which function has a domain of all real numbers?

A. y=cos x
B. y=sec x
C. y=1/sin x
D. y=tan x

Please explain. Thank you!

Answer 1

\[\cos(x) = (-\infty,+\infty)\]\[y=\sec(x) \implies y= \frac{1}{\cos(x)};~ \text{domain is restricted by cos(x) }\]\[y=\frac{1}{\sin(x)}; ~\text{domain is restricted by sin(x)}\]\[y=\frac{\sin(x)}{\cos(x)} ;~\text{domain is restricted because of cos(x)}\]

Answer 2

Does that make sense to you @KJ4UTS

Answer 3

\[\frac{1}{\cos(x)} \implies x > 0;~ (0,+\infty)\]\[\cos(x)=\sin(x) \implies x > 0; ~ (0,+\infty)\]

Answer 4

That's what I meant, if that makes any more sense.

Answer 5

Yes, cosine functions are only limited in the range, as \(-1\le y \le 1\)

Answer 6

Same thing with sine functions, as sine functions are cosine functions but are shifted.

Answer 7

So when you've got a fraction... 1 over something, you're restricting the values the sine or cosine function can be.