Notes & theory for this topic

Trigonometry

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1. Use the cosine rule. Give your answers to 3 significant figures.
  1. A triangle has sides a = 7 mm, b = 9 mm and c = 10 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 = 42.8°, B = 60.9°, C = 76.2°
  2. A triangle has sides a = 6 mm, b = 9 mm and c = 12 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 = 0.505 rad, B = 0.813 rad, C = 1.82 rad
  3. A triangle has sides a = 9 mm, b = 13 mm and c = 14 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).
    8.13 mm
2. Solve the following triangles (angles in radians). Give your answers to 3 significant figures.
  1. In a triangle, A = 0.77 rad, b = 6 and c = 5. 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 = 4.23, B = 1.41 rad, C = 0.965 rad
  2. In a triangle, A = 1.33 rad, b = 13 and c = 15. Find side a and the angles B and C (radians).
    Show answer
    Find a with the cosine rule, then B with the cosine (or sine) rule, and C = π − A − B.
    a = 17.3, B = 0.815 rad, C = 0.997 rad
  3. In a triangle, A = 0.4 rad, B = 1.1 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 = 4.29, b = 9.83
  4. In a triangle, A = 0.6 rad, B = 0.95 rad and the included side c = 13. 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 = 7.34, b = 10.6
3. Convert these Cartesian coordinates to polar form [r, θ] (−π ≤ θ ≤ π), to 3 significant figures.
  1. Convert the Cartesian point (7, 9) to polar coordinates [r, θ], with θ in radians.
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    r = √(x2+y2), θ = atan2(y, x) (mind the quadrant).
    [11.4, 0.910]
  2. Convert the Cartesian point (-7, 9) to polar coordinates [r, θ], with θ in radians.
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    r = √(x2+y2), θ = atan2(y, x) (mind the quadrant).
    [11.4, 2.23]
  3. Convert the Cartesian point (10, -5) to polar coordinates [r, θ], with θ in radians.
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    r = √(x2+y2), θ = atan2(y, x) (mind the quadrant).
    [11.2, -0.464]
  4. Convert the Cartesian point (-9, -11) to polar coordinates [r, θ], with θ in radians.
    Show answer
    r = √(x2+y2), θ = atan2(y, x) (mind the quadrant).
    [14.2, -2.26]
4. Convert these polar coordinates to Cartesian form (x, y), to 3 significant figures.
  1. Convert the polar point [15, 2.09] (θ in radians) to Cartesian coordinates (x, y).
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    x = r·cosθ, y = r·sinθ.
    (-7.44, 13.0)
  2. Convert the polar point [20, 1.92] (θ in radians) to Cartesian coordinates (x, y).
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    x = r·cosθ, y = r·sinθ.
    (-6.84, 18.8)
  3. Convert the polar point [14, 0.46] (θ in radians) to Cartesian coordinates (x, y).
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    x = r·cosθ, y = r·sinθ.
    (12.5, 6.22)
  4. Convert the polar point [20, -1.14] (θ in radians) to Cartesian coordinates (x, y).
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    x = r·cosθ, y = r·sinθ.
    (8.35, -18.2)
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