find the exact solutions in the interval [0,2pi), sin4x=-2sin2x
sin4x=-2sin2x
2 sin2xcos2x = -2sin2x, ......... since sin2A = 2sinAcosA
cos2x = -1
2x = 180°, 360°, 540°, 720° which is π , 2π 3π, or 4π radians
x = 90° ,180°,270°, 360° which is π/2, π , 3π/2, or 2π radians
To find the exact solutions in the interval [0, 2pi) for the equation sin4x = -2sin2x, we can rearrange the equation to a more manageable form.
Let's start by using the double-angle identity for sine: sin2x = 2sinx*cosx. This identity expresses sin2x in terms of sinx and cosx.
Now, let's substitute this identity into the equation sin4x = -2sin2x:
sin4x = -2(2sinx*cosx)
Expanding further, we have:
sin4x = -4sinx*cosx
Next, let's use the double-angle identity for cosine: cos2x = cos^2x - sin^2x. This identity expresses cos2x in terms of cosx and sinx.
Let's substitute this identity into the equation sin4x = -4sinx*cosx:
sin4x = -4sinx*(cos^2x - sin^2x)
Expanding further, we have:
sin4x = -4sinx*cos^2x + 4sin^3x
Now, we can simplify the equation by moving all terms to one side:
sin4x + 4sinx*cos^2x - 4sin^3x = 0
To solve this equation, we can factor out sinx:
sinx * (sin4x + 4cos^2x - 4sin^2x) = 0
Now, we can set each factor equal to zero and solve for x.
1. sinx = 0
In the interval [0, 2pi), sinx = 0 has solutions at x = 0 and x = pi.
2. sin4x + 4cos^2x - 4sin^2x = 0
This equation involves trigonometric functions and solving it directly for exact solutions can be challenging. However, we can simplify it using trigonometric identities.
Using the identity sin^2x + cos^2x = 1, we can rewrite the equation as:
sin4x + 4(1 - sin^2x) - 4sin^2x = 0
Simplifying further, we have:
sin4x + 4 - 4sin^2x - 4sin^2x = 0
Combine like terms:
sin4x - 8sin^2x + 4 = 0
Factoring out sin^2x:
sin^2x(4cos^2x - 7) = 0
Now, we have two possibilities:
a. sin^2x = 0
In this case, sinx = 0 as well, which is already covered in the first solution (sinx = 0).
b. 4cos^2x - 7 = 0
Rearrange this equation:
4cos^2x = 7
Divide by 4:
cos^2x = 7/4
Taking the square root of both sides:
cosx = ±sqrt(7)/2
Now, we can find the values of x that satisfy these solutions. Taking the inverse cosine (cos^-1) of both sides:
x = cos^-1(sqrt(7)/2) or x = cos^-1(-sqrt(7)/2)
Using a calculator to find the inverse cosine of these values, we get:
x ≈ 0.437, 2.705
So, the solutions in the interval [0, 2pi) for the equation sin4x = -2sin2x are:
x = 0, pi, 0.437, 2.705