Well we can apply this formula.
sin x = cos(90-x)
So sin 30 = cos(90-30) = cos 60
sin 30-cos 60
=cos 60 - cos 60
- nooo, thats a blunder. sin30-cos60= sin(90-60)-cos60= cos60-cos60=0 since sin(90-x)=cosx
No please. It is not correct to calculate sin30 - cos60 as 30 - 60. The concept is totally different.
If ABC is a right angled triangle with right angle at B, then sine and cosine ratios are defined as below:
sin A = Opposite angle / hypotenuse = BC/AC
cosC = Adjascent angle/hypotenuse = BC/AC.
Therefore in a right angled triangle if A = x, then C = (90-x) .
Therefore sin A = sinx
sinx = BC/AC...(1).
cosC = cos(90-x).
cos(90-x) = BC/AC....(2)
Therefore from (1) and (2) we get:
sinx = cos(90-x)....(3).
If we put x= 30 in eq (3), we get:
sin30 = cos(90-30) = cos60.
Therefore, sin30 - cos60 = 0..
Therefore sin30 -cos60 = 0 and it is not 30 -60.
You want to calculate the difference of the results of functions sine and cosine and not the difference between their arguments.
The functions sine and cosine don't have the behavior of the identically function, where f(x) = x.
Let's see how it works!
In a right angle triangle, where one cathetus is b and the other one is c and the hypothenuse is a, a cathetus opposite to the angle 30 (meaning pi/6 radians) is half from hypothenuse.
If b is the opposite cathetus to pi/6 angle, that means that b=a/2. In this way, we can find the other cathetus length, using Pythagorean theorem.
a^2=b^2 + c^2
a^2 = a^2/4 + c^2
a^2 - a^2/4 = c^2
3a^2/4 = c^2
sin pi/6=opposite cathetus/hypotenuse
sin pi/6= (a/2)/a
cos pi/3=cos 60=adjacent cathetus/hypotenuse
cos pi/3= (a/2)/a
sin pi/6 - cos pi/3=1/2 -1/2=0
Note the difference!
30 - 60 = -30 = 330 degrees and sin 30 - cos 60 = 0.