Solve sin2x + sin4x = cos2x + cos 4x for x in the interval[0,2pi)
Hint: the following substitution should come in handy: sin3x= cos3x . tan3x
sin2x + sin4x = 2sin3x cosx
cos2x + cos4x = 2cos3x cosx
so, we have
sin3x cosx = cos3x cosx
tan3x = 1
so, 3x = pi/4 or 5pi/4
x = pi(1+4k)/12 for all 0<=k<=5
also x=k*pi/2 (where cosx = 0)
cos4x-cos2x=sin4x-sin2x
To solve the equation sin(2x) + sin(4x) = cos(2x) + cos(4x) for x in the interval [0, 2pi), we will use the given substitution sin(3x) = cos(3x) * tan(3x). Here's how to approach the problem:
Step 1: Express everything in terms of sin(3x)
Let's replace cos(3x) with sin(3x) / tan(3x) in the equation:
sin(2x) + sin(4x) = cos(2x) + (sin(3x) / tan(3x))
Step 2: Simplify the right side of the equation
Combine the terms on the right side of the equation, keeping in mind that tan(3x) = sin(3x) / cos(3x):
sin(2x) + sin(4x) = cos(2x) + (sin(3x) * cos(3x)) / sin(3x)
Step 3: Cancel out sin(3x)
Since sin(3x) appears in both the numerator and denominator of the right side, we can cancel it out:
sin(2x) + sin(4x) = cos(2x) + cos(3x)
Step 4: Express sin(4x) and cos(3x) in terms of sin(2x)
Using the double angle identity sin(2θ) = 2sin(θ)cos(θ), we can express sin(4x) and cos(3x) in terms of sin(2x):
sin(2x) + 2sin(2x)cos(2x) = cos(2x) + cos(3x)
Step 5: Simplify the equation
Combine terms on both sides of the equation:
3sin(2x)cos(2x) = cos(3x)
Step 6: Use the trigonometric identity sin(2θ) = 2sin(θ)cos(θ)
By substituting sin(2x) = 2sin(x)cos(x), we get:
6sin(x)cos^2(x) = cos(3x)
Step 7: Divide by cos^2(x)
Since cos(x) ≠ 0, we can divide both sides of the equation by cos^2(x):
6sin(x) = cos(3x) / cos^2(x)
Step 8: Use the trigonometric identity tan(θ) = sin(θ) / cos(θ)
Replace cos(3x) / cos^2(x) with cot(x), so we have:
6sin(x) = cot(x)
Step 9: Rearrange the equation
Multiply both sides of the equation by cos(x) to get rid of sin(x) on the left side:
6sin(x)cos(x) = cot(x)cos(x)
6sin(x)cos(x) = 1
Step 10: Use the trigonometric identity 2sin(θ)cos(θ) = sin(2θ)
Apply the double angle identity to 6sin(x)cos(x):
3sin(2x) = 1
Step 11: Solve for sin(2x)
Divide both sides of the equation by 3 to isolate sin(2x):
sin(2x) = 1/3
Step 12: Solve for x
To solve sin(2x) = 1/3, we need to find the values of x in the interval [0, 2π] where sin(2x) equals 1/3.
Use the inverse sine function (sin^(-1)) to find the values of x:
2x = sin^(-1)(1/3)
x = (1/2) * sin^(-1)(1/3)
Using a calculator, find sin^(-1)(1/3) ≈ 0.3398.
Finally, substitute the value of sin^(-1)(1/3) into the expression for x:
x ≈ (1/2) * 0.3398 ≈ 0.1699
Therefore, the approximate value for x is around 0.1699 in the interval [0, 2π].