# Which is the value of x for sinx=cosx?

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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

pi/4=45 degrees.

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.

2*x=180-90

x=90/2

**x=45 degrees**

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).