Question

How do you express `f(theta)=2sin(theta/2)-5cos(-theta/2-pi)` in terms of trigonometric functions of a whole theta?

Answer 1

`2 sqrt ( 1/2 ( 1 - cos theta )) - 5 (+- sqrt ( 1/2(1 + cos theta )))`,
choosing `( - )` of `( +- )`, when
#theta in ( 2kpi - pi/2, 2kpi }, k = 1, 2, 3,

Explanation:

`f ( theta ) = 2 sin ( 1/2theta )- 5 cos ( -1/2theta - pi )`

`= 2 sin ( 1/2theta ) - 5 cos ( 1/2theta + pi )`

`= 2 sin ( 1/2theta ) + 5 cos ( 1/2theta )`

`= 2 sqrt ( 1/2 ( 1 - cos theta )) - 5 (+- sqrt (1/2( 1 + cos theta )))`,

choosing `( - )` of `( +- )`, when

#theta in ( 2kpi - pi/2, 2kpi }, k = 1, 2, 3, ... .

I measure #theta as positive, in both anticlockwise and clockwise

rotations.

The reader can realize that

if #theta in ( - pi/2, 0), theta/2 in ( - pi/4, 0 ).

the ambiguity in sign of

`cos (theta/2) = +- sqrt ( 1 + cos theta )`

does not arise at all.

Mathematical exactitude is needed, for disambiguation.

Alone, I am an advocate of strict adherence to'' `r >= 0` and so is

`theta`''.