Which is the value of x for sinx=cosx?
For the beginning, we could give a simple answer: x=pi/4.
Let's solve the equation:
sin x - cos x=0
We'll divide the equation by (cos x) and we'll get:
sin x /cos x - 1 =0
The ratio sinx/cosx = tanx
tan x -1 =0
We'll add 1 both sides:
tan x = 1
x= arctan 1 + k*pi
x = pi/4 + k*pi
We could also make the remark that if the ratio sin x / cos x=1, the terms from numerator and denominator are equal => sin x = cosx.
If sin x = cos x => the angles of the triangle are equal, too, so, in a right angle triangle, the angles could only be = 45 degrees (the conclusion is based on fact that in a triangle, the sum of angles is 180 degrees, and one of them is 90 degrees and the other 2 are equals, so 90 + 2*x=180.
sinx = cosx.
To solve for x.
We know that cos^2x = 1-sin^2x.
so sinx = sqrt(1-sin^2x). Square both sides:
sin^2x = 1-sin^2x
2sin^2 x = 1
sinx ^2 = 1/2
sin x = + or - sqrt(1/2).
x = arc sinx = 45 deg when sinx = cosx = sqrt(1/2)
x = 180+45 = 225 deg when sinx = cosx = - sqrt(1/2).