1. Problem 21: Find the length of side AC in a right triangle ABC with \( \angle A = 90^\circ \), \( \angle B = 60^\circ \), and \( \angle C = 30^\circ \), given \( AB = 6 \) cm. 2. In a 30-60-90 right triangle, the sides are in ratio \(1 : \sqrt{3} : 2\) opposite to angles 30°, 60°, and 90° respectively. 3. Since \( AB = 6 \) cm is opposite \( \angle C = 30^\circ \), \( AB \) corresponds to the shorter leg (opposite 30°), so the hypotenuse \( BC = 2 \times AB = 12 \) cm. 4. Side \( AC \) is opposite \( \angle B = 60^\circ \), so \( AC = AB \times \sqrt{3} = 6 \sqrt{3} \) cm. 5. Hence, \( AC = 6 \sqrt{3} \) cm. 6. Now check each option: A: \( \frac{9\sqrt{3}}{2} + 3 = 4.5\sqrt{3} + 3 \neq 6\sqrt{3} \) B: \( 3 + \frac{9}{2} = 3 + 4.5 = 7.5 \neq 6\sqrt{3} \approx 10.39 \) C: \( \frac{6\sqrt{3}}{2} + \frac{9}{2} = 3\sqrt{3} + 4.5 \approx 10.69 \) close but includes an extra term. D: \( \frac{6\sqrt{3}}{2} + \frac{9\sqrt{3}}{2} = 3\sqrt{3} + 4.5\sqrt{3} = 7.5\sqrt{3} \approx 12.99 \neq 6\sqrt{3} \) 7. None seem exact but the correct length is \( 6 \sqrt{3} \) cm, which matches none exactly. Possibly the correct expression is different or simplified. ----- 8. Problem 22: Given \( \tan x = \frac{1}{\sqrt{3}} \), find \( \sin y \) in right triangle ABC with right angle at B and angles x at C and y at A. 9. Since \( \tan x = \frac{1}{\sqrt{3}} \), \( x = 30^\circ \) because \( \tan 30^\circ = \frac{1}{\sqrt{3}} \). 10. Triangle angles: \( x + y + 90^\circ = 180^\circ \) so \( y = 60^\circ \). 11. Therefore, \( \sin y = \sin 60^\circ = \frac{\sqrt{3}}{2} \). ----- 12. Problem 23: Given \( \sin A = \frac{1}{3} \), find \( \cos A \). 13. Using Pythagorean identity: \( \sin^2 A + \cos^2 A = 1 \). 14. Substitute \( \sin A = \frac{1}{3} \): $$\left(\frac{1}{3}\right)^2 + \cos^2 A = 1$$ $$\frac{1}{9} + \cos^2 A = 1$$ $$\cos^2 A = 1 - \frac{1}{9} = \frac{8}{9}$$ 15. Taking positive root (assuming angle A in first quadrant): $$\cos A = \frac{\sqrt{8}}{3} = \frac{2\sqrt{2}}{3}$$ 16. Checking options, D matches \( \frac{\sqrt{8}}{3} \). Final answers: 21: \( AC = 6\sqrt{3} \) cm (no exact option given) 22: \( \sin y = \frac{\sqrt{3}}{2} \) (Option C) 23: \( \cos A = \frac{\sqrt{8}}{3} \) (Option D)