Question

8sin^2x(3-2sin^2x)-9=0

Answer 1

$8sin^2(x)*(3-2sin^2(x))-9=0$
$-(4sin^2(x)-3)^2=0$
$(4sin^2(x)-3)^2=0$
$4sin^2(x)-3=0$
$4sin^2(x)=3$
$sin^2(x)=\frac{3}{4}$
sin(x)=√3/2                                                   sin(x)=-√3/2
x=(-1)^{n} *arcsin(√3/2 )+πn; n∈Z         x=(-1)^{n} *arcsin(-√3/2)+πn; n∈Z
x=(-1)^{n} *π/3 +πn; n∈Z                     x=(-1)^{n} *(-π/3) +πn; n∈Z

Answer 2

To solve the equation 8sin^2(x)(3-2sin^2(x)) - 9 = 0, we can use some algebraic manipulations. Let's go step by step:

1. Distribute the 8 through the parentheses: 8sin^2(x)(3) - 8sin^2(x)(2sin^2(x)) - 9 = 0

2. Simplify: 24sin^2(x) - 16sin^4(x) - 9 = 0

3. Rearrange the terms: -16sin^4(x) + 24sin^2(x) - 9 = 0

4. Let's substitute sin^2(x) with a new variable, let's say u: -16u^2 + 24u - 9 = 0

5. Now, we have a quadratic equation in u. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula: u = (-b ± √(b^2 - 4ac)) / (2a)

Plugging in the values: u = (-24 ± √(24^2 - 4(-16)(-9))) / (2(-16)) u = (-24 ± √(576 - 576)) / (-32) u = (-24 ± √0) / (-32)

6. Simplify the expression: Since the discriminant (√0) is zero, we have only one solution for u: u = -24 / (-32) u = 3 / 4

7. Now, substitute back sin^2(x) for u: sin^2(x) = 3 / 4

8. Take the square root of both sides: sin(x) = ±√(3 / 4)

9. Simplify: sin(x) = ±√3 / 2

10. Solve for x: x = arcsin(√3 / 2) or x = π - arcsin(√3 / 2)

Therefore, the solutions for the equation 8sin^2(x)(3-2sin^2(x)) - 9 = 0 are: x = π/3 + 2πn or x = 2π/3 + 2πn, where n is an integer.