Notes & theory for this topic

Trigonometry

Write your working on paper. Reveal each answer only after you've had a go.

1. Use the cosine rule. Give your answers to 3 significant figures.
  1. A triangle has sides a = 8 mm, b = 11 mm and c = 15 mm. Find the angles A, B and C in degrees.
    Show answer
    Use the cosine rule, e.g. A = cos-1((b2+c2−a2)/2bc). Check the three angles sum to 180°.
    A = 31.3°, B = 45.6°, C = 103°
  2. A triangle has sides a = 13 mm, b = 14 mm and c = 15 mm. Find the angles A, B and C in radians.
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    Use the cosine rule, e.g. A = cos-1((b2+c2−a2)/2bc). Check the three angles sum to π rad.
    A = 0.927 rad, B = 1.04 rad, C = 1.18 rad
  3. A triangle has sides a = 5 mm, b = 7 mm and c = 9 mm. Find the perpendicular height from the largest angle to the side opposite it.
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    Find the area (e.g. by Heron's formula), then height = 2 × area / (longest side).
    3.87 mm
2. Solve the following triangles (angles in radians). Give your answers to 3 significant figures.
  1. In a triangle, A = 1.63 rad, b = 10 and c = 13. Find side a and the angles B and C (radians).
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    Find a with the cosine rule, then B with the cosine (or sine) rule, and C = π − A − B.
    a = 16.9, B = 0.633 rad, C = 0.878 rad
  2. In a triangle, A = 2.02 rad, b = 9 and c = 6. Find side a and the angles B and C (radians).
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    Find a with the cosine rule, then B with the cosine (or sine) rule, and C = π − A − B.
    a = 12.8, B = 0.686 rad, C = 0.436 rad
  3. In a triangle, A = 0.55 rad, B = 1.06 rad and the included side c = 9. Find sides a and b.
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    C = π − A − B, then use the sine rule a/sin A = b/sin B = c/sin C.
    a = 4.71, b = 7.86
  4. In a triangle, A = 0.67 rad, B = 0.79 rad and the included side c = 11. Find sides a and b.
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    C = π − A − B, then use the sine rule a/sin A = b/sin B = c/sin C.
    a = 6.87, b = 7.86
3. Convert these Cartesian coordinates to polar form [r, θ] (−π ≤ θ ≤ π), to 3 significant figures.
  1. Convert the Cartesian point (8, 12) to polar coordinates [r, θ], with θ in radians.
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    r = √(x2+y2), θ = atan2(y, x) (mind the quadrant).
    [14.4, 0.983]
  2. Convert the Cartesian point (-5, 5) to polar coordinates [r, θ], with θ in radians.
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    r = √(x2+y2), θ = atan2(y, x) (mind the quadrant).
    [7.07, 2.36]
  3. Convert the Cartesian point (6, -14) to polar coordinates [r, θ], with θ in radians.
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    r = √(x2+y2), θ = atan2(y, x) (mind the quadrant).
    [15.2, -1.17]
  4. Convert the Cartesian point (-6, -8) to polar coordinates [r, θ], with θ in radians.
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    r = √(x2+y2), θ = atan2(y, x) (mind the quadrant).
    [10.0, -2.21]
4. Convert these polar coordinates to Cartesian form (x, y), to 3 significant figures.
  1. Convert the polar point [15, 2.38] (θ in radians) to Cartesian coordinates (x, y).
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    x = r·cosθ, y = r·sinθ.
    (-10.9, 10.4)
  2. Convert the polar point [6, 0.82] (θ in radians) to Cartesian coordinates (x, y).
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    x = r·cosθ, y = r·sinθ.
    (4.09, 4.39)
  3. Convert the polar point [17, 2.53] (θ in radians) to Cartesian coordinates (x, y).
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    x = r·cosθ, y = r·sinθ.
    (-13.9, 9.76)
  4. Convert the polar point [12, 0.25] (θ in radians) to Cartesian coordinates (x, y).
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    x = r·cosθ, y = r·sinθ.
    (11.6, 2.97)
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