Worksheets for GCSE Mathematics. Trigonometry. Mr Black's Maths Resources for Teachers GCSE 1-9. Shape

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1 Worksheets for GCSE Mathematics Trigonometry Mr Black's Maths Resources for Teachers GCSE 1-9 Shape

2 Pythagoras Theorem & Trigonometry Worksheets Contents Differentiated Independent Learning Worksheets Pythagoras Theorem Finding the Hypotenuse Pythagoras Theorem Finding the Shorter Side Trigonometry - Calculating a Length Trigonometry - Calculating an Angle Trigonometry Mixed Problems Three Dimensional Trigonometry Applying the Sine Rule Applying the Cosine Rule to Calculate Lengths Applying the Cosine Rule to Calculate Angles Area of a triangle formula Trigonometric graphs and equations Sin, Cos & Tan 30, 45, 60 and 90 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Solutions Pythagoras Theorem Finding the Hypotenuse Pythagoras Theorem Finding the Shorter Side Trigonometry - Calculating a Length Trigonometry - Calculating an Angle Trigonometry Mixed Problems Three Dimensional Trigonometry Applying the Sine Rule Applying the Cosine Rule to Calculate Lengths Applying the Cosine Rule to Calculate Angles Area of a triangle formula Trigonometric graphs and equations Sin, Cos & Tan 30, 45, 60 and 90 Page 15 Page 15 Page 15 Page 15 Page 16 Page 16 Page 16 Page 17 Page 18 Page 19 Page 20 Page 21 2

3 Pythagoras Theorem Finding the Hypotenuse Q1. Calculate the length of the hypotenuse for each diagram: Calculate the unknown length in each diagram: Q3. Calculate the length of the line segment between the coordinate pairs. a) (1,2) & (4,6) b) (0,3) & (5,8) c) (-3,3) & (0, 1) d) (-2,-1) & (-4,-3) 3

4 Pythagoras Theorem Finding the Shorter Side Q1. Calculate the length of the unknown side for each diagram: Calculate the unknown length in each diagram: 4

5 Trigonometry - Calculating a Length Q1. Calculate the length of the angle in each right angled triangle. Calculate the unknown length in each diagram: a) A wire 20 metres long runs from the top of a pole to the ground. The wire makes an angle of 40 with the ground. Calculate the height of the pole. Give your answer to 3 significant figures. b) One end of a rope of length 62 metres is tied to a point on the ground and the other end to the top of a mast. When the rope is taut, its inclination to the horizontal is 52. Calculate, to three significant figures the distance of the top of the mast from the ground. 5

6 Trigonometry - Calculating an Angle Q1. Calculate the length of the angle in each right angled triangle. Calculate the unknown angle in each diagram: 6

7 Trigonometry Mixed Problems Q1. Calculate the value of the unknown in each diagram: a) b) i) Calculate the length of CB. ii) Calculate the angle CBD. I) Calculate angle EGF ii) Calculate the length FH c) d) i) Calculate the length TU. ii) Calculate the angle UST. i) Calculate the angle WYX. ii) Calculate the length of ZX. Triangle HLJ and IJK are mathematically similar. Length IJ = 4cm Angle IJK = 62 Calculate the length of KL. 7

8 Q1. Three Dimensional Trigonometry i) Calculate the length AG ii) Calculate the angle AGB. i) Calculate the angle FED ii) Calculate the length FE. iii) Calculate the angle EFI i) Calculate the length XS ii) Calculate the angle XSW. iii) Calculate the length WS. M is the midpoint of line JG. i) Calculate the length ME ii) Calculate the angle ME makes with the plane ELKF. Calculate: a) The length FA b) The angle FEAC makes with the plane ABCD. 8

9 Applying the Sine Rule Q1. Calculate the unknown angle, θ, in each diagram. a) b) c) Calculate the unknown length in each diagram. a) b) c) Calculate length XY. Calculate length GH. Calculate length UR. Q3. a) The triangle ABC has length AC = 10.4 cm, BC = 8.2cm and angle CAB = 51. Calculate angle ABC. b) The triangle ABC has length AB = 3 cm, BC = 1.5 cm and angle CAB = 29. Calculate the ABC. c) The triangle XYZ has length, XY = 10 cm, angle YXZ = 15 and angle XYZ = 36. Calculate the length of side XZ. d) The triangle STU has length ST = 9 cm, angle TSU = 64 and angle SUT = 55. Calculate the length UT. Q4. Two lighthouses track the same ferry. Lighthouse A is 50 Km south of lighthouse B. The ferry is on a bearing of 060 from lighthouse A and 160 from lighthouse B. Calculate the distance the ferry is from both lighthouses. Q5. In triangle ABC, DC = 4, AC = 8, Angle CDB = 50 and angle DBC = 42. a) Calculate angle DAC. b) Calculate the length BA

10 Applying the Cosine Rule to Calculate Lengths Q1. Calculate the unknown lengths in each of these triangles. Calculate the unknown lengths in each of these triangles. a) Triangle ABC has AB = 7cm, AC = 18.3 cm and angle BAC = 57. Calculate the length BC. b) Triangle ABC has BC = 3.8 cm, AB = 4 cm and angle CBA = 70. Calculate the length AC. c) Triangle XYZ has XZ = 0.7 cm, ZY = 0.8 cm and angle XZY = 46. Calculate the length of XY. Q3. Paul and Clare begin an orienteering course from the same point. For the first leg Paul walks 6.4 Km on a bearing of 087. Clare walks on a bearing of 40 for 10.6 Km. Calculate the distance between them for the first leg of the course. Q4. In triangle ABC, side AB = xx + 1, AC = xx and angle CAB = 60. Show that CCCC 2 = xx 2 + xx

11 Cosine Rule Calculating Angles Q1. Calculate the unknown angle in each of these triangles: a) In triangle ABC length AB = 4.5cm, AC = 8cm and BC = 7.2cm. Calculate the angle BAC. b) In triangle XYZ length XZ = 8cm, ZY = 11.7cm and XY = 5cm. Calculate the angle XYZ. c) In triangle PQR length PQ = 4.59cm, QR = 7.28cm and RP = 5.87cm. Calculate the each of the angles in the triangle. Q3. From the triangle ABC: a) Show that CCCCCC αα = 1 3 (xx+5 xx+1 ) b) Given that αα = 60. Calculate the value of xx. Q4. Calculate the angle-marked θ. 11

Area of a Triangle using Trigonometry Q1.

a) b) c) d) e) f ) a) Find the area of triangle ABC if

c) Find the area of an equilateral triangle of length

Find the side length x if the area of the triangle is

Q4) a) Find the area of a triangle ABC if AB = 5 cm,

b) Find the area of a triangle XYZ if XY = 8 cm, XZ =

12 Area of a Triangle using Trigonometry Q1. Calculate the area of the following triangles. a) b) c) d) e) f ) a) Find the area of triangle ABC if AB = 9 cm, AC = 12 cm and CAB = 91 b) Find the area of triangle XYZ if XY = 11.7 cm, YZ = 11.2 cm and XYZ = 55. c) Find the area of an equilateral triangle of length 8 cm. Q3. Find the side length x if the area of the triangle is 60 cm 2. Q4) a) Find the area of a triangle ABC if AB = 5 cm, AC = 11.5 cm and BC = 11 cm. b) Find the area of a triangle XYZ if XY = 8 cm, XZ = 8.2 cm and YZ = 14 cm. c) Find the area of a triangle PQR if PQ = 2.5 cm, QR = 3.5 cm and PR = 4 cm. d) Find the area of a triangle EFG if EF = 6 cm, EG = 14 cm and FG = 9 cm. Q5. Find the perimeter of the triangle if its area is 150 cm 2. 12

13 Q1. For 360 x 360, sketch the graphs of Trigonometric Graphs a) Sin x b) Cos x c) Tan x The diagram shows a sketch of the curve y = sin x for 0 x 360 The exact value of Sin 30 is 0.5. Write down the exact values of a) Sin 150 b) Sin 210 Q3. Give another angle which have a value equal to each of the these between the range 360 x 360. a) Cos 60 b) Sin 270 c) Sin 45 d) Cos 135 e) Sin 60 f) Cos 180 g) Tan 45 h) Tan 60 Q3. For 0 x 360, solve each of these. a) Sin x = 2 2 b) Sin x = 3 2 c) Tan x = 1 d) Sin x = 0.8 e) Tan x = 1.56 f) Cos x = 1 g) Cos x =0 h) Sin x =

a) Sin 30 b) Cos 60 c) Tan 45 d) Cos 30 e) Tan 30 f) Cos 45 g) Sin 60 h) Sin 90 i) Tan 60 j) 4Sin 60 k) 2Cos 30 l) 6Tan 60 m) Cos 45

14 Exact Trigonometric Solutions Q1. Calculate the marked length and angle for each of these special triangles. i) ii) Calculate the exact values of the following trigonometric functions. a) Sin 30 b) Cos 60 c) Tan 45 d) Cos 30 e) Tan 30 f) Cos 45 g) Sin 60 h) Sin 90 i) Tan 60 j) 4Sin 60 k) 2Cos 30 l) 6Tan 60 m) Cos 45 n) 3Tan 30 o) 2Sin 45 Q3. Use the trigonometric graphs to write the following in their exact form: a) Sin 135 b) Tan 225 c) Cos 300 d) Tan 240 e) Cos 330 f) Sin 150 g) 2Sin135 h) Cos Sin 150 i) Cos 360 Tan45 14

15 Solutions Pythagoras Theorem Finding the Hypotenuse Q1 a) 4.47 cm b) 5.83 cm c) 7.62 cm d) 5 cm e) 6.4 cm f) 7.21 cm g) 8.49 cm h) 9.43 cm i) cm a) 7.81 cm b) 5.83 cm c) 5 cm Q3. a) 5 units b) 7.07 units c) 3.61 units d) 2.83 units Pythagoras Theorem Finding the Shorter Side Q1. a) 4.47 cm b) 6.32 cm c) 7.48 cm d) 9.17 cm e) 7.42 cm f) 8.66 cm a) 6.71 cm b) h = 15.2 cm, g = cm c) t = cm u = cm d) k = 8.93 cm Trigonometry - Calculating a Length Q1. a) 5.82 cm b) 4.94 cm c) 3.06 cm d) 6.36 cm e) 3.09 cm f) cm g) 2.27 cm h) 5.83 cm i) 8.69 cm j) 10.5 cm k) cm l) cm a) Height of the pole = 12.9 m b) Distance of the top of the mast from the ground = 48.9 m Trigonometry - Calculating an Angle Q1. a) 31 b) 30 c) 48 d) 71 e) 71 f) 39 g) 37 h) 37 i) 45 j) 57 k) 44 l) 39 a) α = 68 b) γ = 72 15

16 Trigonometry Mixed Problems Q1. a) CB = 3.5 cm, CBD = 59 b) EGF = 45, FH = cm c) TU = cm, UST = d) WYX = 26.57, ZX = 4.17 cm KL = 4.01 cm Three Dimensional Trigonometry Q1 a) AG = 6.4 cm, AGB = b) FED = 18.43, FE = 6.32 cm, EFI = c) XS = 7.81 cm, XSW = 52.01, d) ME = 8.94 cm, MEX = WS = cm FA = cm, Angle between FEAC & ABCD = Applying the Sine Rule Q1. a) θ = 29 b) θ = 46.2 c) θ = 34.7 a) Length XY = 6.5 units b) Length GH = 7.1 units c) Length UR = 5.1 units Q3. a) Angle ABC = 49. b) Angle ABC = 80 c) Length XZ = 13.2 cm d) Length UT = 9.2 cm Q4. Distance from A = 4.4 Km Distance from B = 1.74 Km Q5. Angle DAC = 22.5 Length BC = 10.8 units 16

17 Applying the Cosine Rule to Calculate Lengths Q1. a) 6.24 units b) = 4.1 units c) 5.2 units d) 5.1 units e) = 4.1 units f) 4.9 units a) BC = 15.7 cm b) AC = 3.1 cm c) XY = 1 cm Q3. Q4. 17

18 Cosine Rule Calculating Angles Q1. a) 70.5 b) 34 c) 70.5 d) 82.8 e) 41.4 f) 48.5 a) BAC = 63.2 b) XYZ = 33.2 c) QPR = 87.3 RQP = 53.6, PRQ = 39 Q3. a) b) CCCCCCCC = (xx + 1) xx 2 2 xx (xx + 1) CCCCCCCC = xx2 + 2xx xx 2 6(xx + 1) 2xx + 10 CCCCCCCC = 6(xx + 1) 2(xx + 5) CCCCCCCC = 6(xx + 1) 1(xx + 5) CCCCCCCC = 3(xx + 1) 1 1(xx + 5) = 2 3(xx + 1) 2(xx + 5) 1 = 3(xx + 1) 3xx + 3 = 2xx + 10 xx = 7 Q4. Θ =

19 Solutions Area of a Triangle using Trigonometry Q1. a) 10.4 cm 2 b) 14.8 cm 2 c) 10.5 cm 2 d) 9.5 cm 2 e) 9.6 cm 2 f ) 73.5 cm 2 a) 54 cm 2 b) 53.7 cm 2 c) 27.7 cm 2 Q3. x = 20 cm Q4) a) 27.3 cm 2 b) 28.5 cm 2 c) 4.3 cm 2 d) 18.4 cm 2 Q cm 19

20 Solutions Trigonometric Graphs Q1. a) 0.5 b) -0.5 Q3. a) -240, -60, 240 b) -90, c) -135, -45, 135 d) -225, -135, 225 e) -120, -60, 120 f) -180 g) -225, -45, 225 h) -300, -120, 60, 240, Q3. a) 45, 135 b) 60, 120 c) 45, 225 d) 53.1, e) 122.7, f) 0, 360 g) 90, 270 h) 23.6,

21 Exact Trigonometric Solutions Solutions Q1. a = 3 b = 1 c = 30 d = 60 e = g = 45 f = 2 a) 1 2 b) 1 2 c) 1 d) 3 2 e) 3 3 f) 2 2 g) 3 2 h) 1 i) 3 j) 2 3 k) 3 l) 6 3 m) 2 2 n) 3 o) 2 Q3. a) 2 2 b) 1 c) 1 2 d) 3 e) 3 2 f) 1 2 g) 2 h) 0 i) 0 21

Mathematics. Geometry. Stage 6. S J Cooper

Mathematics. Geometry. Stage 6. S J Cooper Mathematics Geometry Stage 6 S J Cooper Geometry (1) Pythagoras Theorem nswer all the following questions, showing your working. 1. Find x 2. Find the length of PR P 6cm x 5cm 8cm R 12cm Q 3. Find EF correct