Short Answer 
The problem involves a right triangle ABC where ∠A is 90°, ∠B is 30°, and ∠C is 60°. Using the Pythagorean theorem, side AB is found to be 9√3, and the trigonometric ratios calculated are sin(C) = 2/3, tan(C) = 3, and sin(B) = 2/1.
Step 1: Understand the Triangle and its Angles
In triangle ABC, we have a right angle at ‚à†A, with angles ‚à†B = 30¬∞ and ‚à†C = 60¬∞. The sides are defined as follows: AC = 9 (the side opposite ‚à†B) and BC = 18 (the hypotenuse). It’s important to familiarize yourself with the triangle’s dimension and angle measures as they will be crucial for applying trigonometric ratios.
Step 2: Apply the Pythagorean Theorem
To find the length of side AB, we use the Pythagorean theorem expressed as: hypotenuse² = perpendicular² + base². In our case, this translates to: BC² = AB² + AC². Plugging in the known lengths, we have:
- (18)² = AB² + (9)²
- 324 = AB² + 81
- Solving for AB gives us AB = 9‚àö3.
Step 3: Determine Trigonometric Ratios
Now, we can compute the necessary trigonometric ratios using the values of the sides found:
- sin(C) = opposite/hypotenuse = AC/BC = 9/18 = 2/3.
- tan(C) = opposite/adjacent = AC/AB = 9/(9‚àö3) = 3.
- sin(B) = opposite/hypotenuse = AC/BC = 9/18 = 2/1.
Thus, the calculated ratios are: sin(C) = 2/3, tan(C) = 3, and sin(B) = 2/1.