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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 ($n = 5$): $\text{Exterior Angle} = \frac{360^\circ}{5} = 72^\circ$ To convert it to 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 $4 \cot 45^\circ - \sec^2 60^\circ + \sin 30^\circ$.

We first compute the trigonometric values:

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

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

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

To convert from Cartesian $(x, y)$ to polar $(r, \theta)$, we use:

• $r = \sqrt{x^2 + y^2}$
• $\theta = \tan^{-1}\left(\frac{y}{x}\right)$

Given $(x, y) = (5, 5)$: $r = \sqrt{5^2 + 5^2} = \sqrt{50} = 5\sqrt{2}$ $\theta = \tan^{-1}\left(\frac{5}{5}\right) = \tan^{-1}(1) = 45^\circ = \frac{\pi}{4} \, \text{radians}$ Answer: $\left( 5\sqrt{2}, \frac{\pi}{4} \right)$.

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

We rewrite $125$ as $5^3$, so the equation becomes: $5^{2x + 7} = 5^3$ Since the bases are the same, equate the exponents: $2x + 7 = 3$ Solve for $x$: $2x = 3 - 7 = -4$ $x = -2$ 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 $(x, y)$ where $x \in A$ and $y \in B$: $A \times B = \{(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)\}$ Answer: $\{(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)\}$.

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

The function $(f + g)(x)$ is defined as: $(f + g)(x) = f(x) + g(x)$ Substitute the given functions: $(f + g)(x) = (x^2 + 2) + (2x^2 - 5) = x^2 + 2x^2 + 2 - 5 = 3x^2 - 3$ Answer: $3x^2 - 3$.

Would you like any further details or clarifications on any of these problems?

Here are 5 related questions:

1. How do you calculate the exterior angle of an octagon?
2. What is the general formula for the cotangent of an angle?
3. How do you convert polar coordinates to Cartesian coordinates?
4. What happens if $5^{2x+7} = 625$?
5. Can you describe the properties of Cartesian products?

Tip: When solving trigonometric expressions, always ensure you're working in the correct unit (degrees or radians).