How do you solve sin^2(x/3) - cos^2 (x/3) = 0?

Question

How do you solve sin^2(x/3) - cos^2 (x/3) = 0?

Answer 1

put sin^2(x/3)=t

then cos^2(x/3)=1-t

t-1+t=0

2t=1

t=1/2

sin^2(x/3)=1/2

sin(x/3)=sqrt(2)/2

x/3=pi/4 and 3pi/4

x=3pi/4+2k*pi and pi/4+2k*pi

I hope you understand this and good luck!

Answer 2

Hi,

sin² (x/3) + cos² (x/3) = 1 is a Pythagorean Identity.

sin² (x/3) - cos² (x/3) = 0

------------------------------------- Add these together.

2sin² (x/3) = 1

sin² (x/3) = ½

sin (x/3) = ±1/√2

x/3 = 45° ± 90k°, where k is any integer

x = 135° ± 270k°, where k is any integer <==ANSWER

I hope that helps!! :-)

Answer 3

[sin(x/3)-Cos(x/3)][sin(x/3)+cos(x/3)]=0

If sin(x/3)-cos(x/3)=0 then sin(x/3)=cos(x/3)  and we have :

x/3=pi/4 or x=3pi/4

x/3=5pi/4 or x=15pi/4=3pi/4

If sin(x/3)+cos(x/3)=0 then sin(x/3)=-cos(x/3) and we have:

x/3=3pi/4 or x=9pi/4=pi/4

x/3=7pi/4 or x=21pi/4=pi/4

So the answers are 2kpi+pi/4 and 2kpi+ 3pi/4

Answer 4

Use trig identity: 1- sin^2(x/3) = cos^2 (x/3) to get

sin^2(x/3) -(1-sin^2(x/3))=0

2sin^2(x/3)-1= 0

sin^2(x/3)=1/2

sin(x/3) = (sqrt2)/2 or -(sqrt2)/2

x/3 = arcsin (sqrt2)/2or arcsin -(sqrt2)/2

x= 3 pi/4 or -3pi/4

Answer 5

that will be,

sin^2(x/3) - [ 1 - sin^2(x/3) ] = 0 ------------ [since sin^2(x) + cos^2(x) = 1]

=> 2sin^2(x/3) -1 = 0

=> sin^2(x/3) = 1/2 = sin^2(pi/4) ------------ [since sin(pi/4) = 1/root2 ]

=> x/3 = pi/4

=> x = 3pi/4

thts it.. :)

Answer 6

that will be,

sin^2(x/3) - [ 1 - sin^2(x/3) ] = 0

[ sin^2(x) + cos^2(x) = 1]

=> 2sin^2(x/3) -1 = 0

=> sin^2(x/3) = 1/2 = sin^2(pi/4)

[ sin(pi/4) = 1/root2 ]

=> x/3 = pi/4

=> x = 3pi/4

Answer 7

Four solutions: x = x = 3Pi/4, Pi/4, 7Pi/4 and 5Pi/4

Here is the calculus:

We know that for any angle:

sin^2(t) + cos^2 (t) = 1

We can substitute in the first equation using the angle = x/3

We obtain that: sin^2(x/3) - 1 + sin^2 (x/3) = 0

That is:

sin^2(x/3)  = 1/2

Sin (2x/3) = 1/root square (2)

we have two solutions:

Sin (2x/3) = + 1/root square (2)

Sin (2x/3) = - 1/root square (2)

the angles which sin is equal to that are:

a) 45º (or Pi/4)

b) 135º (or 3Pi/4)

c) 225º (or 5Pi/4)

d) 315º (or 7Pi/4)

So the values of x are those that complain with: x/3 = angles described above

x = 3Pi/4, Pi/4, 7Pi/4 and 5Pi/4

Hope it helps.