Title MSTE 7 Solutions.pdf ✅

07 Trigonometry: Trigonometric Functions Solutions ▣ 1. Find the amplitude and period of y = sin x cos x. [SOLUTION] From double angle identity, sin 2x = 2 sin x cos x sin x cos x = 1 2 sin 2x y = 1 2 sin 2x The period is P = 2π 2 = π ▣ 2. If sin 3x = cos 9y, what is x + 3y equal to? [SOLUTION] From trigonometric identities, sin x = cos(90° − x) sin x = cos y → x + y = 90° Therefore, if sin 3x = cos 9y, 3x + 9y = 90 x + 3y = 30° ▣ 3. From point A, at street level and 205 ft from the base of a building, the angle of elevation to the top of the building is 23.1°. Also, from point A the angle of elevation to the top of a neon sign, which is at the top of the building is 25.9°. Determine the height of the building. [SOLUTION]
H = 205 tan 23.1° H = 87.44 ft ▣ 4. In the previous question, how tall is the neon sign? [SOLUTION] h = 205 tan 25.9° − 205 tan 23.1° h = 12.10 ft ▣ 5. Given a triangle ABC, how many possible triangle/s can be formed from the following conditions? AB = 18, AC = 25, C = 42°. [SOLUTION] From cosine law, AB 2 = AC 2 + BC 2 − 2(AC)(BC) cos C 182 = 252 + x 2 − 2(25)x cos 42° x = 25.22, 11.93 Since BC has 2 possible lengths, then there are 2 possible triangles that can be formed.
▣ 6. Given two sides and an angle of a triangle ABC: AB = 40 cm, AC = 35 cm, B = 65°, how many distinct triangle/s can be formed? [SOLUTION] From cosine law, AC 2 = AB 2 + BC 2 − 2(AB)(BC) cos B 352 = 402 + x 2 − 2(40)(x) cos 65° x = 16.9047 ± 9.446i Since both values are imaginary, then there are 0 triangles. ▣ 7. Points A and B 1000 m apart are plotted in a straight highway running East and West. From A, the bearing of a tower C is N 32° W and from B, the bearing of C is N 26° E. Compute the shortest distance of tower C to the highway. [SOLUTION] From the figure, x tan 26° + x tan 32° = 1000 x = 898.794 m ▣ 8. From a window O of a tall building 92 m away above a level ground, the angle of depression of the top of a nearby tower is 38°. From the base of the building the angle of elevation of the top of the tower is 22°. Find the height of the tower. [SOLUTION]
h + d tan 38° = 92 d = h tan 22° h + h tan 22° tan 38° = 92 h = 31.3592 m ▣ 9. In the previous question, find the distance from the tower to the building. [SOLUTION] d = h tan 22° d = 31.3592 tan 22° d = 77.6167m ▣ 10. In question 8, find the angle subtended by the tower from point O. [SOLUTION] tan(θ + 38°) = 92 d tan(θ + 38°) = 92 77.6167 θ = 11.847°