Here is the breakdown and solution for each of the questions:

Q3: Find the degree and radian measure of the exterior angle of a regular pentagon.

• A regular polygon's exterior angle is given by: $\text{Exterior angle} = \frac{360^\circ}{n}$ For a pentagon, $n = 5$: $\text{Exterior angle} = \frac{360^\circ}{5} = 72^\circ$ Now, converting to radians: $\text{Radians} = 72^\circ \times \frac{\pi}{180^\circ} = \frac{2\pi}{5} \text{ radians}$ Answer: $72^\circ$ or $\frac{2\pi}{5}$ radians.

Q4: Evaluate $4 \cot 45^\circ - \sec^2 60^\circ + \sin 30^\circ$.

• $\cot 45^\circ = 1$,
• $\sec 60^\circ = 2$, so $\sec^2 60^\circ = 4$,
• $\sin 30^\circ = \frac{1}{2}$.

Now substitute: $4 \cot 45^\circ - \sec^2 60^\circ + \sin 30^\circ = 4(1) - 4 + \frac{1}{2} = 4 - 4 + \frac{1}{2} = \frac{1}{2}$ Answer: $\frac{1}{2}$.

Q5: Find the polar coordinates of the point whose Cartesian coordinates are $(5, 5)$.

• To convert from Cartesian to polar coordinates, we use the formulas: \quad \text{and} \quad \theta = \tan^{-1} \left(\frac{y}{x}\right)$$ Here, $$x = 5$$, $$y = 5$$: $$r = \sqrt{5^2 + 5^2} = \sqrt{50} = 5\sqrt{2} \quad \text{and} \quad \theta = \tan^{-1} \left(\frac{5}{5}\right) = \tan^{-1}(1) = 45^\circ = \frac{\pi}{4} \text{ radians}$$ **Answer:** $$\left(5\sqrt{2}, \frac{\pi}{4}\right)$$.

Q6: Solve $5^{2x+7} = 125$.

• Express 125 as a power of 5: $125 = 5^3$.
• Now, equate the exponents: $5^{2x+7} = 5^3$ This gives: $2x + 7 = 3$ Solving for $x$: \quad \Rightarrow \quad x = -2$$ **Answer:** $$x = -2$$.

Q7: If $A = \{1, 2, 3\}$ and $B = \{a, b\}$, find $A \times B$.

• The Cartesian product $A \times B$ consists of all ordered pairs where the first element is from set $A$ and the second element is from set $B$. $A \times B = \{(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)\}$ Answer: $\{(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)\}$.

Q8: If $f(x) = x^2 + 2$ and $g(x) = 2x^2 - 5$, then find $(f + g)(x)$.

• Add the two functions: $(f + g)(x) = f(x) + g(x)$ Substituting: $(f + g)(x) = (x^2 + 2) + (2x^2 - 5) = x^2 + 2x^2 + 2 - 5 = 3x^2 - 3$ Answer: $(f + g)(x) = 3x^2 - 3$.

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Follow-up Questions:

1. How do you convert degrees to radians for any angle?
2. What is the general formula for the exterior angle of any polygon?
3. How do you find the Cartesian product of two sets?
4. Can you express $\cot$, $\sec$, and $\sin$ in terms of sine and cosine?
5. What are the steps to solve exponential equations involving powers of the same base?

Tip:

When solving trigonometric problems, it's helpful to remember exact values for common angles (e.g., 30°, 45°, 60°).