Trigonometry

56.
The diagram shows a radio station tower which is built in two sections.
The angle of elevation of the top of the first section is 25° and for the top of the second section is 40°. AB = 10 m.
A B C D 25° 40° 10 m NOT TO SCALE

(a) Find the angle of depression of B from D.
......................... [1]
(b) Calculate:
(i) BC,
......................... m [2]
(ii) The angle B ˆ D C ,
......................... [2]
(iii) DC.
......................... m [2]
Paper 3
98.
Points C and G lie on horizontal line HE as illustrated in the diagram.
Lines BH and DG are vertical.
BC=80 cm, HC=60 cm, GE=35 cm, DG=40 cm, and angle DCG=32°.
NOT TO SCALE 32° 80 cm 60 cm 40 cm 35 cm B H C D G E

Use any of the information in the table where necessary.
sin−1 (34) = 48.6° tan32° = 0.623 cos−1 (78) = 29.0°
cos−1 (34) = 41.4° sin32° = 0.530 tan−1 (78) = 41.2°
tan−1 (34) = 36.9° cos32° = 0.848 sin−1 (78) = 61.0°

(a) Calculate:
(i) The size of the angle HBC
(ii) The length of line CD, leaving the answer as a fraction
(b) Write the angle of depression of C from D.
(a)(i) ......................... [1] (a)(ii) ......................... [1] (b) ......................... [3]
Paper 1
100.
ABC is a triangle in which AC = 12 cm and BC = 9 cm as shown in the diagram.
12 cm 9 cm A B C

(a) Find AB.
(b) Write the value of angle BÂC.
(a) ......................... [2] (b) ......................... [2]
Paper 1
102.
In the triangle ABC, AB=12 cm, AC=5 cm and angle AB∠=θ.
NOT TO SCALE θ 5 cm 12 cm A B C

Find:
(a) BC
(b) the value of cos⁡θ
(a) ......................... (b) .........................
Paper 1
105.
In the diagram, angle ABC=90°, BC=8 cm and AC=10 cm.
NOT TO SCALE θ 10 cm 8 cm A B C

(a) Calculate AB
(b) Write the value of cos⁡θ in its simplest form.
(a) ......................... (b) .........................
Paper 1
57.
The diagram shows triangles ABC and ADE. The straight line ACD is perpendicular to both AB and DE.
AB = 12 cm, BC = 13 cm, CD = 11 cm and ∠ D A E = 40 ° .
B A C D E 12 13 11 40° NOT TO SCALE

Calculate:
(a) The obtuse angle BCD,
......................... [3]
(b) The distance AE.
......................... cm [3]
Paper 3
58.
The diagram shows a ladder, AC, leaning against a house, BCEFGH.
AB = 5 m, BC = 3 m, CE = 2 m and ∠ E C D = 30 ° .
A B C D E F G H θ 5 m 3 m 2 m 30° NOT TO SCALE

(a) Calculate the length:
(i) AB,
......................... m [2]
(ii) ED.
......................... m [2]
(b) Find the height of the house.
......................... m [2]
(c) Calculate the angle θ .
......................... [2]
Paper 3
59.
The diagram shows triangle ABC. BD is a straight line and D is a point on AC.
Angle ADB = 110°, angle ABD = 42° and BD = BC.
A D C B 110° 42° NOT TO SCALE

(a) What is the special name for the triangle BCD?
......................... [1]
(b) Find the size of angle:
(i) BAD,
......................... [2]
(ii) ABC.
......................... [2]
(c) It is given that BD = 7 cm. Calculate the shortest distance from point D to AB.
......................... cm [3]
(d) The area of triangle ABD is 38 cm2. Calculate the length AB.
......................... cm [2]
Paper 3
60.
(a) The diagram shows triangle ABC. Angle ABC = 90°, AB = 6.4 cm and BC = 8 cm.
A B C 6.4 cm 8 cm NOT TO SCALE

Calculate:
(i) AC,
......................... cm [2]
(ii) Angle CAB.
......................... [2]
(b) In triangle LMN, angle LMN = 90°, LM = 8 cm and LN = 12.3 cm.
(i) Calculate angle MLN.
......................... [2]
(ii) Given that L is due north of M, calculate the bearing of L from N.
......................... [2]
(iii) MN is extended and the point P lies on this extended line so that LP = 17.6 cm. Find NP.
......................... cm [3]
Paper 3
61.
The diagram shows a quadrilateral ABCD.
A B C D 16 cm 12 cm 40° NOT TO SCALE

AB = 16 cm, AD = 12 cm.
Angle BCD = 40°, angle ADB = angle CBD = 90°.
Calculate the length of CD. Give your answer correct to 3 significant figures.
CD = ......................... cm [5]
Paper 3