Let's solve the questions in the uploaded image step by step with explanations.

Question 2: Completions

(a) Simplify the trigonometric expression:

$\sin 30^\circ + 2\cos 60^\circ - \tan^2 45^\circ$ Solution:

• $\sin 30^\circ = \frac{1}{2}$,
• $\cos 60^\circ = \frac{1}{2}$,
• $\tan 45^\circ = 1$.

Substitute these values: $\sin 30^\circ + 2\cos 60^\circ - \tan^2 45^\circ = \frac{1}{2} + 2 \cdot \frac{1}{2} - 1^2$ $= \frac{1}{2} + 1 - 1 = \frac{1}{2}.$

Final Answer: $\frac{1}{2}$.

(b) Find $C$, the midpoint of $AB$, where $A(1, 5)$ and $B(3, 7)$.

Formula for the midpoint: $C(x, y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right).$

Substitute $A(1, 5)$ and $B(3, 7)$: $C = \left(\frac{1 + 3}{2}, \frac{5 + 7}{2}\right) = \left(\frac{4}{2}, \frac{12}{2}\right) = (2, 6).$

Final Answer: $C(2, 6)$.

(c) Find the slope of $AB$, where $A(2, -3)$ and $B(1, -5)$:

Formula for slope: $m = \frac{y_2 - y_1}{x_2 - x_1}.$

Substitute $A(2, -3)$ and $B(1, -5)$: $m = \frac{-5 - (-3)}{1 - 2} = \frac{-5 + 3}{1 - 2} = \frac{-2}{-1} = 2.$

Final Answer: Slope $m = 2$.

(d) Solve for $\cos C \cdot \tan A$ in triangle $ABC$, where:

• Right triangle, $\angle B = 90^\circ$,
• $AB = 5$, $BC = 12$.

Find $AC$ using the Pythagorean theorem: $AC = \sqrt{AB^2 + BC^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13.$

$\cos C = \frac{BC}{AC} = \frac{12}{13}$, and $\tan A = \frac{BC}{AB} = \frac{12}{5}$.

$\cos C \cdot \tan A = \frac{12}{13} \cdot \frac{12}{5} = \frac{144}{65}$.

Final Answer: $\frac{144}{65}$.

(e) Solve for $m$, where $\tan(x + 25^\circ) = 1$ and $x + 25^\circ$ is acute:

If $\tan(x + 25^\circ) = 1$, then: $x + 25^\circ = 45^\circ.$

Solve for $x$: $x = 45^\circ - 25^\circ = 20^\circ.$

Final Answer: $x = 20^\circ$.

(f) Find the coordinates of the center of a circle where $AB$ is its diameter:

• $A(-1, -4)$ and $B(7, 6)$.

The center is the midpoint of $AB$: $\text{Midpoint} = \left(\frac{-1 + 7}{2}, \frac{-4 + 6}{2}\right) = \left(\frac{6}{2}, \frac{2}{2}\right) = (3, 1).$

Final Answer: Center at $(3, 1)$.

Do you want me to continue solving the next questions or provide more detailed explanations for any part?

Here are 5 related questions:

1. How do you derive the midpoint formula geometrically?
2. Why does $\tan(x) = 1$ imply specific angle values?
3. Can you calculate the slope if $x_1 = x_2$? What happens?
4. How do you confirm if three points form a right triangle?
5. How does the Pythagorean theorem extend to coordinate geometry?

Tip: Always double-check trigonometric simplifications for precision in larger problems!