1. Convert 3\pi radians to grads. Since $$1\text{ radian} = \frac{200}{\pi} \text{ grads}$$, multiply: $$3\pi \times \frac{200}{\pi} = 600 \text{ grads}$$. None of the options exactly match 600 grads, so the closest provided option is not correct based on conversion. 2. Convert 3200 mils to degrees. Since $$1 \text{ mil} = \frac{360}{6400} = 0.05625^\circ$$, Calculate: $$3200 \times 0.05625 = 180^\circ$$. Answer: a. 180° 3. Find the third leg in the triangle with given sides 7, 4, and 3.6. Assuming it's a right triangle and 3.6 is one leg, 4 is another leg, find the hypotenuse: $$\sqrt{7^2 - 4^2} = \sqrt{49 - 16} = \sqrt{33} \approx 5.744$$ Given options are different; if 7 and 4 are legs, Calculate the third side using Pythagoras: $$\sqrt{7^2 + 4^2} = \sqrt{49 +16} = \sqrt{65} \approx 8.06$$ Closest option is b. 8.46. 4. Given $$\sin \theta = \frac{3}{5}$$ for acute angle $$\theta$$, find $$\cos \theta$$ using Pythagorean identity: $$\cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5}$$. Answer: a. ⅘ 5. Right triangle sides 7 cm, 24 cm, and 25 cm (hypotenuse). Find angle opposite 7 cm side: $$\sin \theta = \frac{7}{25}$$, $$\theta = \arcsin \left( \frac{7}{25} \right) \approx 16.26^\circ$$. Closest option: b. 16.3° 6. Given $$\sec x = 14.6401$$, find $$x$$: $$\sec x = \frac{1}{\cos x}$$, $$\cos x = \frac{1}{14.6401} \approx 0.0683$$, $$x = \arccos(0.0683) \approx 86.08^\circ$$. Answer: a. 86.083 7. Angle of elevation to building top is 30°, to flag top is 45°. Building height: 10 m. Let distance from P to building base be $$d$$. $$\tan 30^\circ = \frac{10}{d} \Rightarrow d = \frac{10}{\tan 30^\circ} = \frac{10}{0.577} = 17.32 m$$. $$\tan 45^\circ = \frac{10 + h}{d} = 1 = \frac{10 + h}{17.32}$$, Solve for $$h$$: $$10 + h = 17.32 \Rightarrow h = 7.32 m$$. Answer: d. 7.32 m 8. Angle to top of building is 28°. Moving 150 ft closer, angle is 42°. Let height be $$h$$, initial distance $$x$$. $$\tan 28^\circ = \frac{h}{x}$$ and $$\tan 42^\circ = \frac{h}{x - 150}$$. From first, $$h = x \tan 28^\circ = x \times 0.5317$$. From second, $$h = (x - 150) \times 0.9004$$. Set equal: $$x \times 0.5317 = (x - 150) \times 0.9004$$, $$0.5317 x = 0.9004 x - 135.06$$, $$0.9004 x - 0.5317 x = 135.06$$, $$0.3687 x = 135.06$$, $$x = \frac{135.06}{0.3687} = 366.42 ft$$. Find $$h$$: $$h = 366.42 \times 0.5317 = 194.8 ft$$. Answer: b. 194.8 ft 9. Bird atop 12m pole, student 20 m away. Find angle $$\theta$$: $$\tan \theta = \frac{12}{20} = 0.6$$, $$\theta = \arctan(0.6) \approx 31^\circ$$. Answer: a. 31° 10. Ladder reaches 6 m height, inclined at 60°. Distance from foot of ladder to wall is: $$\text{base} = \text{ladder length} \times \cos 60^\circ$$. Ladder length = hypotenuse = 6 / sin 60° = $$6 / 0.866 = 6.93 m$$. Calculate base: $$6.93 \times 0.5 = 3.464 m$$. Answer: a. 3.464 m