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Trigonometry Sine Area Rule

B. Trigonometry Sine Area Rule. a. C. c. b. By Mr Porter. A. Definition: The area of any triangle, ABC, can be found by using the formula:. B. b. c. C. A. a. The formula is cyclic:. Example 1: Calculate the area of triangle ABC, correct to 3 sig. fig.

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Trigonometry Sine Area Rule

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  1. B TrigonometrySine Area Rule a C c b By Mr Porter A

  2. Definition: The area of any triangle, ABC, can be found by using the formula: B b c C A a The formula is cyclic:

  3. Example 1: Calculate the area of triangle ABC, correct to 3 sig. fig. Example 2: Calculate the area of triangle ABC, correct to 1 dec. pl.. r 12.5 cm Q P 108°40’ B Label the sides of the triangle a, b and c.. 13.8 cm Label the side of the triangle p, q and r. p b q c 55 m R 42°15’ Write the the formula for this triangle. C A 62 m a Write the the formula for this triangle. Substitute the values for p, r and angle Q. Substitute the values for a, b and angle C. Evaluate and round to 1 dec. pl. Evaluate and round to 3 sig. fig.

  4. Example 3: Calculate the area of quadrilateral, correct to 3 sig. fig. Side c is missing! (Hypotenuse of ∆ABD)  Pythagoras’ Thm. c2 = a2 + b2 C Note: To find the area of the quadrilateral, you need to calculate the area of the 2 triangles. c2 = 202 + 152 c2 = 625 c =√625 c = 25 Side BD is 25 m. B 35 m 15 m 55°30’ m A D Area of ∆BDC, sine area formula 20 m Label Quadrilateral (Triangles) Area of triangle ABD, AABD = 1/2 x B x H AABD = ½ x 20 x 15 AABD = 150 m2 Area of ∆BDC, sine area formula Hence, the total area = 150 + 360.555 = 510.555 = 511 m2

  5. Example 4: Calculate the area of regular hexagon of radius 12 m, correct to 3 sig. fig. Example 5: Calculate the area of regular pentagon of radius 15 cm, correct to 3 sig. fig. Draw a diagram with radial diagonals. Draw a diagram with all radii from the centre drawn. Angle at the centre for a regular polygon is θ = 360÷ number of sides. Angle at the centre for a regular polygon is θ = 360÷ number of sides. 12 m 15 cm 12 m θ = 60° 15 cm 72° θ = 72° 60° 12 m 15 cm 12 m 15 cm Area is 6 times the area of 1 triangle. Area is 5 times the area of 1 triangle. Area of ∆ABC, sine area formula Area of ∆ABC, sine area formula Substitute the values for a, b and angle C. Substitute the values for a, b and angle C.

  6. Example 6: The results of a radial survey are shown in the diagram (all measurements in metres). Calculate the total area of ∆XYZ (nearest m2 ). Now apply the sine area rule 3 times. Area of ∆XOZ: North (0°T) X (045°T, 68m) Likewise, areas for ∆XOY and ∆YOZ O Z (275°T, 92 m) Area of ∆XOY= 3258.854 m2 Area of ∆YOZ= 4063.391 m2 Y (155°T, 102m) Hence, total area = 2396.187 + 3258.854 + 4063.391 = 9718.432 m2 = 9718 m2 Need to calculate the area of all 3 triangles. Must calculate the angles around ‘O’. From their bearing: Angle XOY = 155 – 45 = 110° Angle YOZ = 275 – 155 = 120° Now, Angle XOZ = (360 – 275) + 45 = 130° OR, Angle XOZ = 360 – (110 + 120) = 130°

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