Sure, let's tackle these questions one step at a time:
### Task 2:
#### (i) Factorize [tex]\(3 x^2 - 12x + 9\)[/tex]
1. Identify coefficients: The quadratic expression is [tex]\(3x^2 - 12x + 9\)[/tex]. Here, [tex]\(a = 3\)[/tex], [tex]\(b = -12\)[/tex], and [tex]\(c = 9\)[/tex].
2. Find the greatest common divisor (GCD): First, observe that each term is divisible by 3. Factor out the 3:
[tex]\[
3(x^2 - 4x + 3)
\][/tex]
3. Factorize the quadratic inside the parentheses: Next, factorize the quadratic [tex]\(x^2 - 4x + 3\)[/tex]:
[tex]\[
x^2 - 4x + 3 = (x - 3)(x - 1)
\][/tex]
4. Combine the factored form: Putting it all together, we get:
[tex]\[
3(x - 3)(x - 1)
\][/tex]
So, [tex]\(3x^2 - 12x + 9\)[/tex] factorizes to [tex]\(3(x - 3)(x - 1)\)[/tex].
#### (ii) Factorize [tex]\(3 \sin^2 \theta - 12 \sin \theta + 9\)[/tex]
1. Identify coefficients: The trigonometric expression is [tex]\(3 \sin^2 \theta - 12 \sin \theta + 9\)[/tex], which is analogous to the quadratic form from part (i).
2. Factor out the GCD: Since this is similar in structure to our previous quadratic form, we factor out the 3:
[tex]\[
3(\sin^2 \theta - 4 \sin \theta + 3)
\][/tex]
3. Factorize the quadratic inside the parentheses: Next, factorize the quadratic [tex]\(\sin^2 \theta - 4 \sin \theta + 3\)[/tex]:
[tex]\[
\sin^2 \theta - 4 \sin \theta + 3 = (\sin \theta - 3)(\sin \theta - 1)
\][/tex]
4. Combine the factored form: Putting it together, we get:
[tex]\[
3(\sin \theta - 3)(\sin \theta - 1)
\][/tex]
So, [tex]\(3 \sin^2 \theta - 12 \sin \theta + 9\)[/tex] factorizes to [tex]\(3(\sin \theta - 3)(\sin \theta - 1)\)[/tex].
#### (iii) Solve the trigonometric equation [tex]\(3 \sin^2 \theta - 12 \sin \theta + 9 = 0\)[/tex] in the interval [tex]\([0, 4 \pi]\)[/tex]
1. Set each factor to zero: From our factorized form [tex]\(3 (\sin \theta - 3)(\sin \theta - 1) = 0\)[/tex], set each factor to zero:
[tex]\[
\sin \theta - 3 = 0 \quad \text{or} \quad \sin \theta - 1 = 0
\][/tex]
[tex]\[
\sin \theta = 3 \quad \text{or} \quad \sin \theta = 1
\][/tex]
2. Solve for [tex]\(\theta\)[/tex]:
- [tex]\(\sin \theta = 3\)[/tex]: The sine function ranges between -1 and 1, so this equation has no real solutions.
- [tex]\(\sin \theta = 1\)[/tex]: This equation has solutions where the angle corresponding to [tex]\(\theta\)[/tex] gives a sine value of 1.
3. Identify the angles:
- [tex]\(\sin \theta = 1\)[/tex] at [tex]\(\theta = \frac{\pi}{2} + 2k\pi\)[/tex], where [tex]\(k \in \mathbb{Z}\)[/tex].
4. List the solutions in the specified interval [tex]\([0, 4\pi]\)[/tex]:
[tex]\[
\theta = \frac{\pi}{2}, \frac{5\pi}{2}, \frac{9\pi}{2}
\][/tex]
So, the solutions to the equation [tex]\(3 \sin^2 \theta - 12 \sin \theta + 9 = 0\)[/tex] in the interval [tex]\([0, 4\pi]\)[/tex] are:
[tex]\[
\theta = \frac{\pi}{2}, \frac{5\pi}{2}, \frac{9\pi}{2}
\][/tex]
### Task 3:
Please provide the details of Task 3 for further assistance.
