Topic Review — Trigonometry — Solutions

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1. Naming sides and setting up ratios

   1. All three ratios: sin(47°) = 9.3/12.7  •  cos(47°) = 8.7/12.7  •  tan(47°) = 9.3/8.7
   2. Which ratio:
      1. sine
      2. cosine
      3. tangent
   3. Equation for opposite: sin(31°) = x/18
   4. Equation for hypotenuse: cos(65°) = 7/x

2. Find the unknown side

   1. 36°, H=20, find opposite: 20×sin(36°) = 20×0.5878 = 11.76
   2. 54°, H=15, find adjacent: 15×cos(54°) = 15×0.5878 = 8.82
   3. 22°, A=10, find opposite: 10×tan(22°) = 10×0.4040 = 4.04
   4. 68°, O=24, find hypotenuse: 24÷sin(68°) = 24÷0.9272 = 25.88
   5. 43°, A=17, find hypotenuse: 17÷cos(43°) = 17÷0.7314 = 23.24
   6. 75°, O=11, find adjacent: 11÷tan(75°) = 11÷3.7321 = 2.95

3. Find the unknown angle

   1. O=5, H=13: θ = sin⊃⁻¹(5/13) = sin⊃⁻¹(0.3846) = 22.6°
   2. A=7, H=25: θ = cos⊃⁻¹(7/25) = cos⊃⁻¹(0.28) = 73.7°
   3. O=9, A=12: θ = tan⊃⁻¹(9/12) = tan⊃⁻¹(0.75) = 36.9°
   4. O=3.6, H=8.4: θ = sin⊃⁻¹(3.6/8.4) = sin⊃⁻¹(0.4286) = 25.4°
   5. O=A=14: θ = tan⊃⁻¹(1) = 45.0° (no calculator needed)
   6. Sides 8 and 15; hypotenuse = √(64+225) = 17: θ⊂1; = tan⊃⁻¹(8/15) = 28.1°; θ⊂2; = 90 − 28.1 = 61.9°

4. Elevation, depression, and bearings

   1. Tree; dist=65 m, elev=38°: h = 65×tan(38°) = 65×0.7813 = 50.78 m
   2. Drone; ht=55 m, elev=62°: tan(62°) = 55/d → d = 55÷tan(62°) = 55÷1.8807 = 29.24 m
   3. Tower; ht=45 m, dep=19°: tan(19°) = 45/d → d = 45÷0.3443 = 130.70 m
   4. Boat bearing 070°, 90 km: Eastward = 90×sin(70°) = 90×0.9397 = 84.57 km. Northward = 90×cos(70°) = 90×0.3420 = 30.78 km.

5. Multi-step problems

   1. Zip-line; ht=18 m, dep=35°:
      1. Length = 18÷sin(35°) = 18÷0.5736 = 31.38 m
      2. Horizontal = 18÷tan(35°) = 18÷0.7002 = 25.71 m
   2. Flagpole from both sides: Height = 25×tan(52°) = 25×1.2799 = 32.00 m. Distance B = 32.00÷tan(38°) = 32.00÷0.7813 = 40.96 m
   3. Roof truss; base = 9 m, angles = 28°:
      1. Half-base = 4.5 m (adjacent). Height = 4.5×tan(28°) = 4.5×0.5317 = 2.39 m
      2. Slope = 4.5÷cos(28°) = 4.5÷0.8829 = 5.10 m
   4. Pilot; alt=3000 m, dep A=12°, dep B=28°: Distance to A = 3000÷tan(12°) = 3000÷0.2126 = 14 110 m. Distance to B = 3000÷tan(28°) = 3000÷0.5317 = 5642 m. Between landmarks = 14 110 − 5642 = 8468 m ≈ 8.47 km