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 = 9 mm, b = 13 mm and c = 14 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 = 38.7°, B = 64.6°, C = 76.7°
  2. A triangle has sides a = 7 mm, b = 10 mm and c = 12 mm. Find the angles A, B and C in radians.
    Show answer
    Use the cosine rule, e.g. A = cos-1((b2+c2−a2)/2bc). Check the three angles sum to π rad.
    A = 0.622 rad, B = 0.984 rad, C = 1.54 rad
  3. A triangle has sides a = 5 mm, b = 9 mm and c = 10 mm. Find the perpendicular height from the largest angle to the side opposite it.
    Show answer
    Find the area (e.g. by Heron's formula), then height = 2 × area / (longest side).
    4.49 mm
2. Solve the following triangles (angles in radians). Give your answers to 3 significant figures.
  1. In a triangle, A = 0.63 rad, b = 13 and c = 12. 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 = 7.80, B = 1.38 rad, C = 1.13 rad
  2. In a triangle, A = 1.19 rad, b = 8 and c = 5. 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 = 7.70, B = 1.30 rad, C = 0.647 rad
  3. In a triangle, A = 0.86 rad, B = 0.61 rad and the included side c = 6. Find sides a and b.
    Show answer
    C = π − A − B, then use the sine rule a/sin A = b/sin B = c/sin C.
    a = 4.57, b = 3.45
  4. In a triangle, A = 0.72 rad, B = 0.77 rad and the included side c = 12. Find sides a and b.
    Show answer
    C = π − A − B, then use the sine rule a/sin A = b/sin B = c/sin C.
    a = 7.94, b = 8.38
3. Convert these Cartesian coordinates to polar form [r, θ] (−π ≤ θ ≤ π), to 3 significant figures.
  1. Convert the Cartesian point (6, 10) to polar coordinates [r, θ], with θ in radians.
    Show answer
    r = √(x2+y2), θ = atan2(y, x) (mind the quadrant).
    [11.7, 1.03]
  2. Convert the Cartesian point (-9, 9) to polar coordinates [r, θ], with θ in radians.
    Show answer
    r = √(x2+y2), θ = atan2(y, x) (mind the quadrant).
    [12.7, 2.36]
  3. Convert the Cartesian point (13, -11) to polar coordinates [r, θ], with θ in radians.
    Show answer
    r = √(x2+y2), θ = atan2(y, x) (mind the quadrant).
    [17.0, -0.702]
  4. Convert the Cartesian point (-14, -7) to polar coordinates [r, θ], with θ in radians.
    Show answer
    r = √(x2+y2), θ = atan2(y, x) (mind the quadrant).
    [15.7, -2.68]
4. Convert these polar coordinates to Cartesian form (x, y), to 3 significant figures.
  1. Convert the polar point [11, 1.75] (θ in radians) to Cartesian coordinates (x, y).
    Show answer
    x = r·cosθ, y = r·sinθ.
    (-1.96, 10.8)
  2. Convert the polar point [13, 0.38] (θ in radians) to Cartesian coordinates (x, y).
    Show answer
    x = r·cosθ, y = r·sinθ.
    (12.1, 4.82)
  3. Convert the polar point [7, -2.12] (θ in radians) to Cartesian coordinates (x, y).
    Show answer
    x = r·cosθ, y = r·sinθ.
    (-3.65, -5.97)
  4. Convert the polar point [15, -1.05] (θ in radians) to Cartesian coordinates (x, y).
    Show answer
    x = r·cosθ, y = r·sinθ.
    (7.46, -13.0)
New questions