Finding Unknown Sides Using Trigonometry — Solutions

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1. Find the value of x

   1. 30°, hyp = 10, find opposite: x = 10 × sin(30°) = 10 × 0.5 = 5.00 cm
   2. 45°, hyp = 8, find adjacent: x = 8 × cos(45°) = 8 × 0.7071 = 5.66 cm
   3. 60°, adj = 5, find opposite: x = 5 × tan(60°) = 5 × 1.7321 = 8.66 cm
   4. 25°, hyp = 14, find opposite: x = 14 × sin(25°) = 14 × 0.4226 = 5.92 cm
   5. 52°, hyp = 9, find adjacent: x = 9 × cos(52°) = 9 × 0.6157 = 5.54 cm
   6. 38°, adj = 11, find opposite: x = 11 × tan(38°) = 11 × 0.7813 = 8.59 cm
   7. 70°, opp = 20, find hypotenuse: x = 20 ÷ sin(70°) = 20 ÷ 0.9397 = 21.28 cm
   8. 15°, adj = 7, find hypotenuse: x = 7 ÷ cos(15°) = 7 ÷ 0.9659 = 7.25 cm
   9. 48°, opp = 6, find adjacent: x = 6 ÷ tan(48°) = 6 ÷ 1.1106 = 5.40 cm
   10. 33°, hyp = 22, find opposite: x = 22 × sin(33°) = 22 × 0.5446 = 11.98 cm
   11. 57°, adj = 13, find hypotenuse: x = 13 ÷ cos(57°) = 13 ÷ 0.5446 = 23.87 cm
   12. 42°, opp = 9.5, find hypotenuse: x = 9.5 ÷ sin(42°) = 9.5 ÷ 0.6691 = 14.20 cm

2. Two-step problems

   1. 34°, adjacent = 16 cm:
      1. Hypotenuse: opposite the right angle. Opposite: across from 34°. Adjacent: 16 cm, between 34° and right angle.
      2. cosine: cos(34°) = adjacent/hypotenuse = 16/x
      3. x = 16 ÷ cos(34°) = 16 ÷ 0.8290 = 19.30 cm
   2. 61°, hypotenuse = 30 cm:
      1. Opposite = 30 × sin(61°) = 30 × 0.8746 = 26.24 cm
      2. Adjacent = 30 × cos(61°) = 30 × 0.4848 = 14.54 cm
      3. Check: 26.24² + 14.54² = 688.54 + 211.41 = 899.95 ≈ 900 ✓
   3. Sam vs Rina: Sam is correct. sin(50°) = opposite/hypotenuse = 8/x, so x = 8/sin(50°) ≈ 10.44. Rina incorrectly placed x on top as if it were the opposite side, giving x = 8×sin(50°) which would only be correct if x were the opposite and 8 the hypotenuse.
   4. 27°, hypotenuse = 18 cm: Opposite = 18 × sin(27°) = 18 × 0.4540 = 8.17 cm (used sine). Adjacent = 18 × cos(27°) = 18 × 0.8910 = 16.04 cm (used cosine).

3. Real-world applications

   1. Tree and shadow:
      1. Right triangle: horizontal shadow (adj = 15 m), vertical tree height (opp = ?), sun angle 42° at the tip of the shadow.
      2. tan(42°) = height / 15
      3. Height = 15 × tan(42°) = 15 × 0.9004 = 13.51 m
   2. Wheelchair ramp:
      1. 0.9 m is the opposite side (vertical rise).
      2. sin(8°) = 0.9/x → x = 0.9 ÷ sin(8°) = 0.9 ÷ 0.1392 = 6.47 m
      3. 6.47 m < 7.5 m, so yes, this ramp complies with the building code.
   3. Cable and pole:
      1. sin(55°) = 12/cable → cable = 12 ÷ sin(55°) = 12 ÷ 0.8192 = 14.65 m
      2. tan(55°) = 12/distance → distance = 12 ÷ tan(55°) = 12 ÷ 1.4281 = 8.40 m
   4. River width:
      1. AB (river width) is the opposite side relative to angle C (62°). AC (40 m walked) is the adjacent side.
      2. tan(62°) = AB / 40 → AB = 40 × tan(62°) = 40 × 1.8807 = 75.23 m

4. Ladder against a wall (72°, base = 2.5 m)

   1. cos(72°) = 2.5/ladder → ladder = 2.5 ÷ cos(72°) = 2.5 ÷ 0.3090 = 8.09 m
   2. tan(72°) = height/2.5 → height = 2.5 × tan(72°) = 2.5 × 3.0777 = 7.69 m
   3. 7.69 m ≥ 7 m, so yes, the ladder complies with the safety guideline.

5. Roof pitch (24°, run = 5 m)

   1. tan(24°) = height/5 → height = 5 × tan(24°) = 5 × 0.4452 = 2.23 m
   2. cos(24°) = 5/rafter → rafter = 5 ÷ cos(24°) = 5 ÷ 0.9135 = 5.47 m

6. Lighthouse (63°, base distance = 18 m)

   1. 18 m is the adjacent side; lighthouse height is the opposite side.
   2. tan(63°) = height/18 → height = 18 × tan(63°) = 18 × 1.9626 = 35.33 m
   3. The opposite side is the lighthouse height (35.33 m); the adjacent side is the distance to the boat (120 m). tan(θ) = 35.33/120 = 0.2944 → θ = tan⊃⁻¹(0.2944) = 16.40° (angle of depression).

7. Ship navigation (bearing N35°E, 80 km)

   1. North displacement = 80 × cos(35°) = 80 × 0.8192 = 65.53 km
   2. East displacement = 80 × sin(35°) = 80 × 0.5736 = 45.89 km

8. Flagpole and shadow (48°, shadow = 9 m)

   1. tan(48°) = height/9 → height = 9 × tan(48°) = 9 × 1.1106 = 9.99 m
   2. New ratio: tan(θ) = 9.99/14 = 0.7136 → θ = tan⊃⁻¹(0.7136) = 35.54°. The sun is lower in the sky.

9. Surveyor and tower (25 m and 40 m positions)

   1. tan(55°) = height/25 → height = 25 × tan(55°) = 25 × 1.4281 = 35.70 m
   2. tan(θ) = 35.70/40 = 0.8926 (this is the value of tan of the new angle of elevation).
   3. θ = tan⊃⁻¹(0.8926) = 41.74°

10. Skatepark ramp design (hypotenuse = 6 m)

    1. Height at 20°: sin(20°) = h/6 → h = 6 × sin(20°) = 6 × 0.3420 = 2.05 m
    2. Height at 30°: sin(30°) = h/6 → h = 6 × sin(30°) = 6 × 0.5 = 3.00 m
    3. sin(θ) = 2.4/6 = 0.4 → θ = sin⊃⁻¹(0.4) = 23.58°. Since 20° ≤ 23.58° ≤ 30°, this angle is within the allowed range.