The diagram shows a right-angled triangle ABC. The right angle is at B, and angle A is marked. AB is the horizontal base, BC is the vertical side, and AC is the sloping side joining A to C.
(a)Which side is the hypotenuse?(1)
(b)Which side is opposite angle A?(1)
(c)Which side is adjacent to angle A?(1)
(Total for Question 1 is 3 marks)
2
A support cable is attached to the top of a flagpole and pegged into the ground. The cable makes an angle of 35 degrees with the ground and is 9 m long. Calculate the height, h, of the flagpole. Give your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 2 is 2 marks)
3
A zip-wire is fixed at the top of a tower and runs down to a point on the ground. The zip-wire is 45 m long and makes an angle of 28 degrees with the horizontal ground. Calculate the horizontal distance, d, from the base of the tower to the point where the zip-wire meets the ground. Give your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 3 is 2 marks)
4
From a point on level ground 18 m from the base of a lighthouse, the angle of elevation to the top of the lighthouse is 52 degrees. Calculate the height of the lighthouse. Give your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 4 is 2 marks)
5
In right-angled triangle DEF, the right angle is at E. EF = 8 cm and the hypotenuse DF = 15 cm. Calculate the size of angle D. Give your answer correct to 1 decimal place.
Diagram NOT accurately drawn
(Total for Question 5 is 2 marks)
6
In right-angled triangle LMN, the right angle is at N. MN = 9 cm is adjacent to angle M, and the hypotenuse ML = 11 cm. Calculate the size of angle M. Give your answer correct to 1 decimal place.
Diagram NOT accurately drawn
(Total for Question 6 is 2 marks)
7
In right-angled triangle XYZ, the right angle is at Y. The side opposite angle X is 7 cm and the side adjacent to angle X is 4 cm. Calculate the size of angle X. Give your answer correct to 1 decimal place.
Diagram NOT accurately drawn
(Total for Question 7 is 2 marks)
8
In right-angled triangle ABC, the right angle is at C. Angle A = 47 degrees and the side AC (adjacent to angle A) = 6.2 cm.
Diagram NOT accurately drawn
(a)Calculate the length of the hypotenuse AB. Give your answer correct to 3 significant figures.(2)
(b)Hence calculate the perimeter of triangle ABC. Give your answer correct to 3 significant figures.(2)
(Total for Question 8 is 4 marks)
9
Priya stands 24 m from the base of a tree, on level ground. The angle of elevation from her eyes to the top of the tree is 31 degrees. Priya's eyes are 1.6 m above the ground. Calculate the total height of the tree. Give your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 9 is 3 marks)
10
From the top of a vertical cliff 52 m high, Tomasz observes a boat out at sea. The angle of depression from Tomasz to the boat is 18 degrees. Calculate the horizontal distance from the base of the cliff to the boat. Give your answer correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 10 is 3 marks)
11
Triangle PQR is isosceles, with PQ = PR = 14 cm and base angles at Q and R both equal to 55 degrees. M is the midpoint of QR, so PM is a line of symmetry perpendicular to QR.
Diagram NOT accurately drawn
(a)Calculate the height PM of the triangle. Give your answer correct to 3 significant figures.(2)
(b)Given that QR = 2 x 14 x cos(55 degrees), calculate the area of triangle PQR. Give your answer correct to 3 significant figures.(3)
(Total for Question 11 is 5 marks)
12
A ship sails from port A on a bearing of 072 degrees for 35 km, reaching point B.
Diagram NOT accurately drawn
(a)Calculate how far east of A point B is. Give your answer correct to 3 significant figures.(2)
(b)Calculate how far north of A point B is. Give your answer correct to 3 significant figures.(2)
(Total for Question 12 is 4 marks)
13
A ladder 5.5 m long leans against a vertical wall, with its foot on horizontal ground. The ladder reaches 4.8 m up the wall.
Diagram NOT accurately drawn
(a)Calculate the angle the ladder makes with the ground. Give your answer correct to 1 decimal place.(2)
(b)Health and safety guidance recommends that a ladder should make an angle of between 70 and 80 degrees with the ground. Using your answer to part (a), determine whether this ladder meets the guidance. Give a reason for your answer.(2)
(Total for Question 13 is 4 marks)
14
A rectangle ABCD has length AB = 12 cm and width BC = 5 cm. The diagonal AC is drawn.
Diagram NOT accurately drawn
(a)Calculate the size of the angle between the diagonal AC and the side AB. Give your answer correct to 1 decimal place.(2)
(b)Hence write down the size of the angle between the diagonal AC and the side BC. Give a reason for your answer.(2)
(Total for Question 14 is 4 marks)
15
A tall building BC stands on horizontal ground, and a vertical flagpole CD stands on top of the building at C. From a point A on the ground, 40 m from the base B of the building, the angle of elevation to the top of the building, C, is 28 degrees, and the angle of elevation to the top of the flagpole, D, is 34 degrees.
Diagram NOT accurately drawn
(a)Calculate the height BC of the building. Give your answer correct to 3 significant figures.(2)
(b)Calculate the height CD of the flagpole. Give your answer correct to 3 significant figures.(3)
(c)Calculate the distance AD, from A to the top of the flagpole. Give your answer correct to 3 significant figures.(2)
(Total for Question 15 is 7 marks)
16
The cross-section of a plastic door-stop is a right-angled trapezium. The top parallel side is 9 cm, the left side is vertical (a right angle at top and bottom), and the slanted right side has length 11 cm and makes an angle of 25 degrees with the horizontal base.
Diagram NOT accurately drawn
(a)Calculate the height of the trapezium and the length of the base. Give each answer correct to 3 significant figures.(3)
(b)Calculate the area of the cross-section.(2)
(Total for Question 16 is 5 marks)
17
This question must be answered without a calculator.
(a)Write down the exact value of cos(30 degrees).(1)
(b)A right-angled triangle has a hypotenuse of length 10 cm and one angle of 30 degrees. Calculate the exact length of the side adjacent to the 30 degree angle. Give your answer in the form a*√3, where a is an integer.(2)
(Total for Question 17 is 3 marks)
18
A vertical tree TB stands on horizontal ground, with its base at B. From a point C on the ground, in line with B, the angle of elevation of the top of the tree, T, is 42 degrees. From a point D, further from the tree than C, with C between B and D and CD = 15 m, the angle of elevation of T is 26 degrees.
Diagram NOT accurately drawn
(a)By forming two equations for the height, h, of the tree (using triangles TBC and TBD), calculate h. Give your answer correct to 3 significant figures.(4)
(b)Hence calculate the distance BC.(2)
(Total for Question 18 is 6 marks)
19
A cuboid ABCDEFGH has a rectangular base ABCD with AB = 8 cm and BC = 6 cm, and vertical edges of height 5 cm (AE = BF = CG = DH = 5 cm). G is the vertex vertically above C. The diagonal AC of the base is drawn, and the space diagonal AG from A to G is also drawn.
Diagram NOT accurately drawn
(a)Calculate the size of the angle between the diagonal AG and the base ABCD.(3)
(b)Calculate the length of the space diagonal AG. Give your answer correct to 3 significant figures.(2)
(Total for Question 19 is 5 marks)
20
Amara, a surveyor, wants to find the width of a river. She stands at point A on one bank, directly opposite a tree T on the opposite bank, so that angle TAB = 90 degrees, where B is a point along her own bank. She walks 60 m along her bank, from A to B, and measures the angle TBA as 38 degrees.
Diagram NOT accurately drawn
(a)Calculate the width of the river, AT. Give your answer correct to 3 significant figures.(2)
(b)Amara checks her result from point C, further along the same bank as B, so that BC = 25 m (with A, B, C in a straight line). Calculate the angle TCA that Amara should expect to measure. She actually measures angle TCA as 29 degrees. Comment on whether her original width, AT, appears to be consistent with this second measurement.(4)
(Total for Question 20 is 6 marks)
Mark scheme · 5.19D Trigonometry (SOHCAHTOA): Fluency and Exam Drill
Question 1
(a) B1 AC cao
(a) Answer: AC
(b) B1 BC cao
(b) Answer: BC
(c) B1 AB cao
(c) Answer: AB
Question 2
M1 h = 9 sin(35) oe
A1 awrt 5.16 (m)
Answer: h = 5.16 m (3 s.f.)
Question 3
M1 d = 45 cos(28) oe
A1 awrt 39.7 (m)
Answer: d = 39.7 m (3 s.f.)
Question 4
M1 height = 18 tan(52) oe
A1 awrt 23 (m)
Answer: height = 23 m (3 s.f.)
Question 5
M1 sin D = 8/15 oe
A1 awrt 32.2 (degrees)
Answer: angle D = 32.2 degrees (1 d.p.)
Question 6
M1 cos M = 9/11 oe
A1 awrt 35.1 (degrees)
Answer: angle M = 35.1 degrees (1 d.p.)
Question 7
M1 tan X = 7/4 oe
A1 awrt 60.3 (degrees)
Answer: angle X = 60.3 degrees (1 d.p.)
Question 8
(a) M1 AB = 6.2 / cos(47) oe
(a) A1 awrt 9.09 (cm)
(a) Answer: AB = 9.09 cm (3 s.f.)
(b) M1 BC = 6.2 tan(47) oe, or use of Pythagoras with their AB (ft)
(b) A1 awrt 21.9 (cm) ft their AB
(b) Answer: perimeter = 21.9 cm (3 s.f.)
Question 9
M1 height above eye level = 24 tan(31) oe
M1 total height = their (24 tan 31) + 1.6
A1 awrt 16 (m)
Answer: total height = 16 m (3 s.f.)
Question 10
M1 correct identification that the angle at the boat (in the right-angled triangle) is 18 degrees, by alternate angles
M1 distance = 52 / tan(18) oe
A1 awrt 160 (m)
Answer: distance = 160 m (3 s.f.)
Question 11
(a) M1 PM = 14 sin(55) oe
(a) A1 awrt 11.5 (cm)
(a) Answer: PM = 11.5 cm (3 s.f.)
(b) M1 QM = 14 cos(55) oe, or QR = 2 x 14 x cos(55)
(b) M1 area = 0.5 x QR x PM (ft their QR and PM)
(b) A1 awrt 92.1 (cm2)
(b) Answer: area = 92.1 cm2 (3 s.f.)
Question 12
(a) M1 east distance = 35 sin(72) oe
(a) A1 awrt 33.3 (km)
(a) Answer: 33.3 km east
(b) M1 north distance = 35 cos(72) oe
(b) A1 awrt 10.8 (km)
(b) Answer: 10.8 km north
Question 13
(a) M1 sin(angle) = 4.8/5.5 oe
(a) A1 awrt 60.8 (degrees)
(a) Answer: 60.8 degrees (1 d.p.)
(b) M1 correct comparison of their angle from part (a) with the range 70 to 80 degrees (ft their angle)
(b) A1 correct conclusion (No) with valid reason, e.g. 60.8 degrees is less than 70 degrees, so the foot of the ladder is too far from the wall / the ladder is at risk of slipping
(b) Answer: No; 60.8 degrees is less than 70 degrees, so the ladder does not meet the guidance.
Question 14
(a) M1 tan(angle) = 5/12 oe
(a) A1 awrt 22.6 (degrees)
(a) Answer: 22.6 degrees (1 d.p.)
(b) M1 90 - their answer to part (a) (ft)
(b) A1 awrt 67.4 (degrees), with reason that angle ABC = 90 degrees so the two parts of the angle sum to 90 degrees
(b) Answer: 67.4 degrees (1 d.p.), because the two angles between AC and the sides AB, BC together make the right angle at B (90 degrees)
Question 15
(a) M1 BC = 40 tan(28) oe
(a) A1 awrt 21.3 (m)
(a) Answer: BC = 21.3 m (3 s.f.)
(b) M1 BD (total height) = 40 tan(34) oe
(b) M1 CD = their BD - their BC (ft)
(b) A1 awrt 5.71 (m)
(b) Answer: CD = 5.71 m (3 s.f.)
(c) M1 AD = 40 / cos(34) oe
(c) A1 awrt 48.2 (m)
(c) Answer: AD = 48.2 m (3 s.f.)
Question 16
(a) M1 height = 11 sin(25) oe
(a) M1 base = 9 + 11 cos(25) oe
(a) A1 height awrt 4.65 (cm) AND base awrt 19 (cm)
(a) Answer: height = 4.65 cm, base = 19 cm (3 s.f.)
(b) M1 area = 0.5 x (9 + their base) x their height, using the trapezium area formula (ft)
(b) A1 awrt 65 (cm2)
(b) Answer: area = 65 cm2 (3 s.f.)
Question 17
(a) B1√3/2 cao
(a) Answer: cos(30 degrees) = √3/2
(b) M1 adjacent = 10 cos(30) oe, using their exact value from part (a)
(b) A1 cao 5*√3 (cm)
(b) Answer: adjacent = 5*√3 cm (= 8.660 cm, awrt 8.66 cm to 3 s.f.)
Question 18
(a) M1 BC = h/tan(42) oe (correct equation from triangle TBC)
(b) M1 AG = √their AC2 + 52 oe (Pythagoras in triangle ACG)
(b) A1 awrt 11.2 (cm)
(b) Answer: AG = 11.2 cm (3 s.f.)
Question 20
(a) M1 AT = 60 tan(38) oe
(a) A1 awrt 46.9 (m)
(a) Answer: AT = 46.9 m (3 s.f.)
(b) M1 AC = AB + BC = 60 + 25 = 85 (m)
(b) M1 angle TCA = tan-1(their AT / 85) oe, using right-angled triangle TAC
(b) A1 awrt 28.9 (degrees)
(b) B1 valid comment comparing the predicted angle (awrt 28.9 degrees) with the measured 29 degrees, e.g. the two values are very close (within 0.2 degrees), so the original width AT = 46.9 m appears consistent, with any small difference likely due to rounding or measurement error
(b) Answer: predicted angle TCA = 28.9 degrees (1 d.p.); since this is very close to the measured 29 degrees, the original measurement of 46.9 m appears consistent.