If sin 2x=cos 3x, solve for x

cos(90-a)=sin a

using this, cos (90-2x)=sin 2x
cos(90-2x)=cos 3x
90-2x=3x
5x=90
x=18

above ans is one of them and not complete ans. Dragon J's answer is correct and complete that is giving all solutions

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Hello,

sin(2x) = cos(3x)

Since sin(θ)=cos(π/2 - θ),
sin(2x) = cos(3x)
cos(π/2 - 2x) = cos(3x)

Hence, with k being any relative integer:
π/2 - 2x = ±3x + 2kπ

Solving the positive equation:
π/2 - 2x = 3x + 2kπ
5x = π/2 + 2kπ
x = π/10 + 2kπ/5 = (x/10)(1 + 4k)
If k=-2, x=-7π/10
If k=-1, x=-3π/10
If k=0, x=π/10
If k=1, x=π/2
If k=2, x=9π/10

Solving the negative equation:
π/2 - 2x = -3x + 2kπ
x = -π/2 + 2kπ

= = = = = = = = = = = = = = = = = = = = = = = =
So the real solutions in radians are of the form:
x = -π/2 + 2kπ
x = π/10 + 2kπ/5 = (π/10)(1 + 4k)

So the real solutions in degrees are of the form:
x = -90° + 360k
x = 18° + 72k

= = = = = = = = = = = = = = = = = = = = = = = =
So the solutions in radians in ]-π; π] are:
x = -7π/10
x = -π/2
x = -3π/10
x = π/10
x = π/2
x = 9π/10

So the solutions in degrees in ]-180; 180] are:
x = -126°
x = -90°
x = -54°
x = 18°
x = 90°
x = 162°

= = = = = = = = = = = = = = = = = = = = = = = =

Methodically,
Dragon.Jade :-)

sin2x = 2(sinx)(cosx)
cos 3x = Cos (2x + x) = Cos2xCosx - Sin2xSinx
Using the formula Cos(A+B) = CosACosB - SinASinB
Cos2X = Cos^2x - Sin^2x = 2Cos^2x - 1 = 1 - 2Sin^2x

Therefore
Cos 3x = (1 - 2Sin^2x)Cosx - (2 SinxCosx)SinX
or Cos 3x = Cosx - 2 Sin^2xCosx - 2 Sin^2xCosx
or Cos 3x = Cos x - 4 Sin^2xCosx
or Cos 3x = Cos x - 4 (1 - Cos^2x)Cosx
or Cos 3x = Cos x - 4 Cos x + 4 Cos^3 x
or Cos 3x = 4 Cos^3 x - 3 Cos x

2(sinx)(cosx) = Cos x - 4 Sin^2xCosx
4 Sin^2xCosx + 2(sinx)(cosx) - Cosx = 0
cosx (4sin^2x + 2sinx -1) = 0
so,
Solve: cosx = 0
and
Solve 4sin^2x + 2sinx -1 = 0
let sinx = A
So, 4A^2 + 2A -1 = 0
To see whether solutions exist, use the determinant
Det = 2^2 - 4 (4)(-1)
Det = 4 + 16 = 20. 20 > 0 so there is more than one solution to 4sin^2x + 2sinx -1 = 0.

Now, use the quadratic formula to solve for A.
4A^2 + 2A -1 = 0
A = cosx = 0.308, -0.809

Now, solve for sinx = 0.308, -0.809 AND as above, cosx = 0

X = 17.9 degrees + 2k(pi), -54 degrees + 2k(pi), 1.57 degrees + 2k(pi)