What is the solution for the equation sinx=cosx?
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sin x - cos x=0
We could divide the equation by (cos x) and the result will be:
sin x /cos x - 1 =0
But we know that the ratio between sinx and cos x determine the tangent function.
tg x -1 =0
We'll move the free term in the right side of the equal sign, like this:
tg x = 1
x= arctg 1 + k*pi
x = pi/4 + k*pi
We could also make the note that if the ratio sin x / cos x=1, it means that the terms from the ratio 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 , given. square both sides and get
(sinx)^2 = ( cosx)^2 =1-(sinx)^2x , as (sinx)^2+cosx)^2 = 1 is a well known identiy .
Rearrange to get, 2(sinx)^2 = 1
Therefore, sinx = +1/sqrt2 or -1/sqrt2
When sinx =1/sqrt2 ,x=45degree or
when sinx -1/sqrt2, x=180+45 = 225 degree.
cos45 = 1/sqrt2 =sin45 and
co225 = -1/sqrt2 =sign225
so X=pi/4(4n+1)..................n belongs to Real numbers
n=0 you get X=45degrees
n=1 you get X=225degrees
put n=-1,-2,-3,....2,3,4,............you got perfact answers
Square both sides of the original equation to get
From the identity `cos^2x=1-(sin^2x)`
we get `sin^2x=1-(sin^2x)`
Rearrange to get `2(sin^2x)=1`
`sinx=1/sqrt(2) or sinx=-1/sqrt(2)`
When `sinx=1/sqrt(2)` ,`x=45^@ or pi/4 rad`
When `sin=-1/sqrt(2)` , `x=225^@ or 5pi/4 rad`
sinx = cosx
tanx = 1
Taking inverse of tan on both sides,
x = 45°
sinx = cosx
divide by cosx :
sinx/cosx = 1
From your own knowledge sinx/cosx is tanx
Tanx = 1
Now tan is positive in the first quarter and the third quarter. To know this you use the square All Station To Central.
The Capitals remind you when the sin , cos, tan are positive.
This square goes anticlockwise. So in the first quarter, all are positive, second only sin, third only tan, fourth only cos.
So tan is positive in the first and third quadrant.
Now take inverse tan on both sides.
this gives x = 45 degrees
Now this is for the first quadrant, to get the angle for the third quadrant you add this to 180 degrees.
Therefore the angle in the third quadrant is 180 + 45 = 225 degrees
Therefore x = 45 degrees or 225 degrees.``
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