# Solve the equation sin x - cos x=0.

jossms | Student, Undergraduate | eNotes Newbie

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To solve sinx-cosx = 0. Since the solution from giorgiana1976 and neela is not the same, one of you must be wrong?

x= pi/4 + k*pi, is not the same as

x= pi/4+k*pi or x= 3pi/4+k*pi

How is that?

neela | High School Teacher | (Level 3) Valedictorian

Posted on

To solve sinx-cosx = 0.

Solution:

Since sin^2x+cos^2=1 is an identity, cosx = (1-sin^2x)^(1/2). So replacing cosx in the given equation we get:

sinx +(1-sin^2)^(1/2) = 0. Or

(1-sin^2x)^(1/2)=-sinx. squaring both sides,

1-sin^2x = sin^2x.Or

1=2sin^2x. Or

sin^2x = 1/2. Or

sinx= sqrt(1/2) or sinx = -sqrt(1/2) .

So x = pi/4 or 3pi/4 Or x= -pi/4 or 5pi/4 Or n*pi/4, where n is an integer.

Or x= 45 degree, 135deg,225 deg or 315deg . or  n*45deg where n is an integer.

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

As we can see,  the equation is linear, in sin x and cos x. The way of solving this kind of equation is to divide enire equation with "cos x".

(sin x/ cos x) - 1 = 0

Instead of the ratio "sin x/ cos x', we'll use the resulting function, tg x.

The obtained equation is an elementary equation.

tg x - 1 = 0

tg x=1

x= arctg 1 +k*pi

x= pi/4 + k*pi