Given 180^ theta 360^, theta is in either the third or fourth quadrant. Since theta = -4sqrt(2)9 is negative, theta must be in the third quadrant, where bo

Here are the solutions to the problems you've provided.

6. Given that $\cos \theta = -\frac{4\sqrt{2}}{9}$, find without using mathematical tables or calculators the value of $\tan \theta$ if $180^\circ \le \theta \le 360^\circ$.

Step 1: Determine the quadrant of $\theta$.
Given $180^\circ \le \theta \le 360^\circ$, $\theta$ is in either the third or fourth quadrant. Since $\cos \theta = -\frac{4\sqrt{2}}{9}$ is negative, $\theta$ must be in the third quadrant, where both sine and cosine are negative.

Step 2: Use the Pythagorean identity to find $\sin \theta$.
The identity is $\sin^2 \theta + \cos^2 \theta = 1$.

$$\sin^2 \theta = 1 - \cos^2 \theta$$ $$\sin^2 \theta = 1 - \left(-\frac{4\sqrt{2}}{9}\right)^2$$ $$\sin^2 \theta = 1 - \frac{16 \times 2}{81}$$ $$\sin^2 \theta = 1 - \frac{32}{81}$$ $$\sin^2 \theta = \frac{81 - 32}{81}$$ $$\sin^2 \theta = \frac{49}{81}$$ $$\sin \theta = \pm\sqrt{\frac{49}{81}} = \pm\frac{7}{9}$$

Step 3: Choose the correct sign for $\sin \theta$.
Since $\theta$ is in the third quadrant, $\sin \theta$ must be negative.

$$\sin \theta = -\frac{7}{9}$$

Step 4: Calculate $\tan \theta$.
The formula for $\tan \theta$ is $\frac{\sin \theta}{\cos \theta}$.

$$\tan \theta = \frac{-\frac{7}{9}}{-\frac{4\sqrt{2}}{9}}$$ $$\tan \theta = \frac{7}{4\sqrt{2}}$$

Step 5: Rationalize the denominator.

$$\tan \theta = \frac{7}{4\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}$$ $$\tan \theta = \frac{7\sqrt{2}}{4 \times 2}$$ $$\tan \theta = \boxed{\frac{7\sqrt{2}}{8}}$$

7. Janet was required to increase a number by 20%. By mistake, she decreased it by 20%. By what percentage should it be increased to give the correct value?

Step 1: Define the original number and the required value.
Let the original number be $N$.
The required value (increase by 20%) is $N + 0.20N = 1.20N$.

Step 2: Define the value Janet obtained by mistake.
The mistake value (decrease by 20%) is $N - 0.20N = 0.80N$.

Step 3: Calculate the percentage increase needed from the mistake value to the required value.
The increase needed is the difference between the required value and the mistake value, divided by the mistake value, then multiplied by 100%.

$$\text{Percentage Increase} = \frac{\text{Required Value} - \text{Mistake Value}}{\text{Mistake Value}} \times 100\%$$ $$\text{Percentage Increase} = \frac{1.20N - 0.80N}{0.80N} \times 100\%$$ $$\text{Percentage Increase} = \frac{0.40N}{0.80N} \times 100\%$$ $$\text{Percentage Increase} = \frac{0.40}{0.80} \times 100\%$$ $$\text{Percentage Increase} = 0.5 \times 100\%$$ $$\text{Percentage Increase} = \boxed{50\%}$$

8. Simplify completely

$$\frac{(2a^3b^5)^4 \div \left(\frac{1}{2}a^3b^2\right)}{(4a^4b^2)^2 \times (8ab)^3}$$

Step 1: Simplify the numerator.
First, simplify $(2a^3b^5)^4$:

$$(2a^3b^5)^4 = 2^4 (a^3)^4 (b^5)^4 = 16a^{12}b^{20}$$

Now, perform the division in the numerator:

$$16a^{12}b^{20} \div \left(\frac{1}{2}a^3b^2\right) = 16a^{12}b^{20} \times \frac{2}{a^3b^2}$$ $$= (16 \times 2) a^{12-3} b^{20-2} = 32a^9b^{18}$$

Step 2: Simplify the denominator.
First, simplify $(4a^4b^2)^2$:

$$(4a^4b^2)^2 = 4^2 (a^4)^2 (b^2)^2 = 16a^8b^4$$

Next, simplify $(8ab)^3$:

$$(8ab)^3 = 8^3 a^3 b^3 = 512a^3b^3$$

Now, perform the multiplication in the denominator:

$$16a^8b^4 \times 512a^3b^3 = (16 \times 512) a^{8+3} b^{4+3} = 8192a^{11}b^7$$

Step 3: Combine the simplified numerator and denominator.

$$\frac{32a^9b^{18}}{8192a^{11}b^7}$$

Step 4: Simplify the expression.
Divide the numerical coefficients:

$$\frac{32}{8192} = \frac{1}{256}$$

Divide the $a$ terms:

$$\frac{a^9}{a^{11}} = a^{9-11} = a^{-2} = \frac{1}{a^2}$$

Divide the $b$ terms:

$$\frac{b^{18}}{b^7} = b^{18-7} = b^{11}$$

Combine these simplified parts:

$$$$