The sum of all the elements in the range of f(x)=( x)+( x)+( x)+( x), x (n )/(2),n Z

Question

The sum of all the elements in the range of $f(x)=(\sin x)+(\cos x)+(	an x)+(\cot x)$, $x
e \frac{n\pi }{2},n\in Z$

Accepted Answer

$f(x)=Sgn(\sin x)+Sgn(\cos x)+Sgn(	an x)+Sgn(\cot x)$

Since the trig functions change signs depending on which quadrant $x$ is in, we analyze the value of $f(x)$ quadrant by quadrant.

Given that $x
e \frac{n\pi }{2}$, we don't have to worry about any of these terms being zero. Each Sgn term will simply be either 1 (if positive) or -1 (if negative).

Quadrant
 Interval
 sin x
 cos x
 tan x
 cot x
 Sum of Sgn values
 f(x)

I
 $0
 +
 +
 +
 +
 1 + 1 + 1 + 1
 4

II
 $\frac{\pi }{2}
 +
 -
 -
 -
 1 - 1 - 1 - 1
 -2

III
 $\pi 
 -
 -
 +
 +
 -1 - 1 + 1 + 1
 0

IV
 $\frac{3\pi }{2}
 -
 +
 -
 -
 -1 + 1 - 1 - 1
 -2

Range of $f(x)$ :

From table, Range $={-2,0,4}$

Sum of all elements in this range $=4+(-2)+0=2$

Solutions · 1