Let’s analyze and solve each question step by step.

1. cos(𝑥) = √3/2, x is acute, then sin(𝑥) = ?

• If cos(𝑥) = √3/2, then 𝑥 = 30°.
• Using the Pythagorean identity:
  $\sin^2(x) + \cos^2(x) = 1$ Substituting $\cos^2(x) = (√3/2)^2 = 3/4$: $\sin^2(x) = 1 - 3/4 = 1/4 \implies \sin(x) = 1/2$
• Answer: (i) 1/2

2. The distance between (3, -5) and the x-axis is...?

• The distance to the x-axis is the absolute value of the y-coordinate:
  $|y| = |-5| = 5$.
• Answer: (iii) 5

3. If tan(𝑥−5) = 1/√3 and (𝑥−5) is acute, then 𝑥 = ?

• If $\tan(𝑥−5) = 1/√3$, then $(𝑥−5) = 30°$.
• Solving for $𝑥$:
  $𝑥 = 30° + 5 = 35°$
• Answer: (i) 35°

4. ABC is a right triangle at B. Sin(A) + Cos(C) = ?

• In a right triangle: \sin(A) = \cos(C) \text{ (since A + C = 90°)}. Therefore: $\sin(A) + \cos(C) = 2\sin(A).$
• Answer: (i) 2sin(A)

5. In a triangle, $\angle A = 85°$, and $\sin(B) = \cos(B)$. Then $\angle C = ?$

• If $\sin(B) = \cos(B)$, then $B = 45°$ (since $\tan(B) = 1$).
• Using the angle sum of a triangle: $\angle C = 180° - \angle A - \angle B = 180° - 85° - 45° = 50°.$
• Answer: (iii) 50°

6. If $x$ is acute and $\sin(x) = 1/2$, then $\sin(2x) = ?$

• Using the double-angle identity:
  $\sin(2x) = 2\sin(x)\cos(x).$ Here, $\sin(x) = 1/2$, and from $\sin^2(x) + \cos^2(x) = 1$, we find: $\cos(x) = √(1 - (1/2)^2) = √(3/4) = √3/2.$ Substituting: $\sin(2x) = 2 \cdot (1/2) \cdot (√3/2) = √3/2.$
• Answer: (iii) √3/2

Let me know if you need more details or explanations for any of these!

Follow-up Questions

1. Can you derive the Pythagorean identity for $\sin(x)$ and $\cos(x)$?
2. How would the answers change if $x$ were obtuse instead of acute?
3. Can you compute distances in 3D space using a similar method as Question 2?
4. What other trigonometric identities can simplify Question 6?
5. How can $\sin(A) + \cos(C)$ change in a non-right triangle?

Tip

Always verify if angles or functions fall in specific quadrants before concluding their values.