Homework Help

What is the solution for the equation sinx=cosx?

user profile pic

pufsipene | Student, Grade 10 | eNoter

Posted April 21, 2009 at 11:46 PM via web

dislike 2 like

What is the solution for the equation sinx=cosx?

7 Answers | Add Yours

user profile pic

giorgiana1976 | College Teacher | Valedictorian

Posted April 24, 2009 at 2:24 AM (Answer #1)

dislike 2 like

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

pi/4=45 degrees.

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

2*x=180-90

x=90/2

x=45 degrees

user profile pic

neela | High School Teacher | Valedictorian

Posted May 27, 2009 at 4:09 PM (Answer #2)

dislike 1 like

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.

*********

Check:

cos45 = 1/sqrt2 =sin45 and

co225 = -1/sqrt2 =sign225

user profile pic

jeyaram | Student, Undergraduate | Valedictorian

Posted September 13, 2009 at 10:26 PM (Answer #3)

dislike 1 like

sinX=cosX

sinX=cos(Pi/2-X)

so  X=pi/4(4n+1)..................n belongs to Real numbers

eg:-

n=0 you get X=45degrees

n=1 you get X=225degrees

put n=-1,-2,-3,....2,3,4,............you got perfact answers

user profile pic

rimmery | Student, Undergraduate | Honors

Posted May 15, 2014 at 2:40 AM (Answer #4)

dislike 0 like

Square both sides of the original equation to get

`sin^2x=cos^2x` 

From the identity `cos^2x=1-(sin^2x)`

we get `sin^2x=1-(sin^2x)`

Rearrange to get `2(sin^2x)=1`

`sin^2x=1/2`

`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` 

user profile pic

maria-vivanco | Student, Grade 11 | Valedictorian

Posted July 30, 2014 at 10:37 PM (Answer #5)

dislike 0 like

45 degrees 

user profile pic

noonaa | Honors

Posted August 1, 2014 at 3:47 PM (Answer #6)

dislike 0 like

sinx = cosx

tanx = 1

Taking inverse of tan on both sides,

x = 45°

user profile pic

user396107 | Honors

Posted September 7, 2014 at 2:47 AM (Answer #7)

dislike 0 like

sinx = cosx

divide by cosx :

sinx/cosx = 1

From your own knowledge sinx/cosx is tanx

therefore:

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.``

``

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes