Find an equation for a periodic function that is always positive. Combine sec x and csc x in such a way that the function crosses the x-axis, and create an expression to describe all of its zeroes.

Question

ive been at this for 3 hours i need help D:

Answer 1

1. Take a periodic function that you know, let's use sin(x). The minimum of sin(x) is -1 , so let f(x)=sin(x)+2 and now the minimum of f(x) is 1 .
2. Seems to me that f(x) = sec(x) + csc(x) does the job.
3. For the zeroes of f(x) = sec(x) + csc(x) :

f(x) = sec(x) + csc(x) = 0

sec(x) = -csx(x)

1/(cos(x)) = -1/(sin(x)) now multiply by sin(x) on both sides

sin(x)/cos(x) = - sin(x)/sin(x)

tan(x) = -1

so x = 3π/4 + π n

where n is any integer because tan(x) = tan(x+π)