Calculate sin 30-cos 60Can i calculate sin30-cos60=30-60 ?
You need to use the following trigonometric identity, such that:
`sin alpha = cos(90^o - alpha)`
Reasoning by analogy, yields:
`sin 30^o = cos (90^o - 30^o) => sin 30^o = cos 60^o`
Moving `cos 60^o` to the left side, yields:
`sin 30^o - cos 60^o = 0`
Hence, evaluating the difference of trigonometric functions, yields `sin 30^o - cos 60^o = 0.`
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.