Trigonometry Topic Question [Pearson Edexcel Mathematics B (4MB1) 2014-2016]

1	(Edexcel/Math B/2015/janR/paper01/q27) A pyramid on horizontal ground has a rectangular base ABCD . Thé vertex \( E \) is vertically above the point \( B \). \( D C .=8 \mathrm{~m} \) and \( B E=12 \mathrm{~m} \). (a) Calculate, in degrees to 3 -significant figures, the angle of elevation of \( E \) from \( A \).[2] The angle of elevation of \( E \) from \( D \) is \( 29^{\circ} \) (b) Calculate the length, in m. to 3 significant figures, of \( A D \).[4]
2	(Edexcel/Math B/2014/june/paper01/q7) The bearing of A from B is 048$^\circ$  Calculate the bearing of B from A.[2]
3	(Edexcel/Math B/2016/june/paper01/q5) The bearing of Nashik from Surat is 142$^\circ$  Calculate the bearing of Surat from Nashik.[2]
4	(Edexcel/Math B/2014/jan/paper01/q30) \( A C D \) is a triangle in which \( \angle A C D=20^{\circ} \) The point \( B \) on \( A C \) is such that \( B D \) is perpendicular to \( C A, \angle B D A=40^{\circ} \) and \( C B=3 \mathrm{~cm} \). The point \( E \) is such that \( A E \) is parallel to \( B D \) and \( D E=2 \mathrm{~cm} \). Calculate the length, in cm.to 3 .significant figures, of (a) \( B D \)[2] (b) \( A D \)[2] (c) Calculate the size, in degrees to 3 significant figures of \(\angle AED \). [3]
5	(Edexcel/Math B/2014/jan/paper01R/q28) \( A, B \) and \( C \) are three points on a horizontal plane. The point \( C \) is due South of the point \( B \) and the point \( A \) is due West of the point \( B \).  A vertical pole, \( B D \), at \( B \) is supported by two cables, \( A D \) and \( C D \).  \( B D=30 \mathrm{~m}, B C=45 \mathrm{~m} \) and \( \angle D A B=40^{\circ} \). Find, giving your answers to 3 significant figures, (a) the length, in m , of the cable \( A D \).[2] (b) the angle of elevation of the point \( D \).from the point \( C \),[2] (c). the bearing of the point \( A \) from the point \( C \).[3]
6	(Edexcel/Math B/2014/jan/paper02R/q6) In Figure 3, \( A \dot{B} C \) is a triangle with \( A B=11 \mathrm{~cm} \) and \( B C=9 \mathrm{~cm} \). The point \( D \) is on \( A C \) such that \( A D=4 \cdot \mathrm{~cm} \) and \( B D=8 \mathrm{~cm} \). Calćulate, to 3 significant figures, (a) the size, in degrees, of \( \angle B D C \),[4] (b) the size, in degrees, of \( \angle B C D \),[3] (c) the area, in \( \mathrm{cm}^{2} \), of triangle \( B D C \).[3] [Cosine rule: \( a^{2}=b^{2}+c^{2}-2 b c \cos A \) Sine rule: \( \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \) Area of a triangle \( \left.=\frac{1}{2} b c \sin A\right] \)
7	(Edexcel/Math B/2014/june/paper01/q23) The diagram shows a right-angled triangle \( A B C \), with \( B C=11 \mathrm{~cm} \) and \( \angle A B C=15^{\circ} \). (a) Calculate the length, in cm to 3 significant figures, of \( A B \).[2]  The point \( D \) on \( A B \) is such that \( A D=A C \). (b) Calculate the ⋅ length, in cm to 3 significant figures, of \( D B \).[3]
8	(Edexcel/Math B/2014/jan/paper01/q1) Figure 1 shows quadrilateral. \( A B C D \) with diagonal \( B D \). Given that \( \angle B A D=110^{\circ}, A B=6 \mathrm{~cm} \) and \( A D=8 \mathrm{~cm} \), (a) calculate the length, in cm to 3 significant figures, of BD.[3]  Givèn also that \( \angle B D C=40^{\circ} \). and \( \angle B C D=60^{\circ} \), calculate the lengtht, in cm to 3 significant figures, of (b) \( B C \),[3] (c) \( A C \).[5]  \[ \left[\begin{array}{c} \text { Sine Rule: } \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \\ \text { Cosine Rule: } a^{2}=b^{2}+c^{2}-2 b c \cos A \end{array}\right] \]
9	(Edexcel/Math B/2014/june/paper01R/q28) The points \( Q, R \) and \( S \) lie in a straight line on horizontal ground with \( Q R=35 \mathrm{~m} \). The vertical mast \( P S \) is such that \( \angle P Q R=25^{\circ} \) and \( \angle P R S=40^{\circ} \) (a) Write down the size, in degrees, of \( \angle Q P R \).[1] (b) Calculate the length, in m to 3 significant figures, of \( P R \).[3] (c) Calculate the height, in \( m \) to 3 significant figures, of the mast PS [2]
10	(Edexcel/Math B/2014/june/paper02R/q10) Figure 2 shows a quadrilateral \( A B C D \cdot \) with \( A B=18 \cdot \mathrm{~m}, D C=24 \mathrm{~m} \) and \( B C=20 \cdot \mathrm{~m} \).  \( \angle A D C=65^{\circ} \) and \( \angle D C B=90^{\circ} \).  Giving all your answers to 3 significant figures, calculate (a) the length, in mi, of \( A C \).[3] (b) the size, in degrees, of \( \angle A C D \)[3] (c) the area, in \( \mathrm{m}^{2} \), of triangle \( A B C \)[3] (d) the area, in \( \mathrm{m}^{2} \), of triangle \( A D B \).[4]  [Cosine rule: \( a^{2}=b^{2}+c^{2}-2 b c \cos A \) Sine rule: \( \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \) ] \[ \text { [Area of triangle } \left.=\frac{1}{2} b c \sin A\right] \]
	to be continued....

Answer Key

 Trigonometry (2014-2016)

1.   (a) $56.3^{\circ} \quad$ (b) $B D=21.6 ; 20.1 \mathrm{~m}$
2.   $ 228^{\circ}$  { or } $ S48W $
3.   $322^{\circ}$
4.   (a)  $B D=1.09 \mathrm{~cm} $ (b) $A D=1.42 ; 1.43 \mathrm{~cm}$ (c) $\angle A E D=27.2,27.3,27.4$
5.   (a) $46.7(\mathrm{~m})$ (B) $ 33.7^{\circ} $ (c) $ 322(321.5)$
6.   (a) $\angle B D C=50.2^{\circ}$ (b) $\angle B C D=43.0^{\circ}$ (c) $35.9 \mathrm{cm}^2$
7.   (a) $ 11.4 \mathrm{~cm}$ (b) $ 8.41,8.44$ or $ 8.45$
8.   (a) $\quad B D=11.5$ (b) $B C=8.55$ or $8.54$ (c) $A C=12.7$
9.   (a) $15^{\circ}$ (b) $57.2 \mathrm{~m}$ (c) $36.7 \mathrm{~m}$
10.   (a) $A C=23.1 $m (b) $\angle A C D=44.9^{\circ} $ (c) $ \triangle A B C=164 \mathrm{~m}^2$ (d) $\triangle A D B=120 \mathrm{~m}^2$
11.   $B C=8.05$
12.   (a) $\angle B A C=43.7^{\circ}$ (b) $\triangle B C Y=19.8$
13.   (a) $O A=11.4$ (b) $A P=8.44$ (c) $B C=4.68$ (d) $B C P Q=10.9 \mathrm{~cm}^2$
14.   $71$
15.   $21 .9 \mathrm{~cm}^2$
16.   (a) 36.2 (b) $47.2 ^\circ$ (c) 155 $cm^2$
17.   $e^\circ=113^\circ$
18.   (a) $A C=3.71$ (b) $\triangle B C D=2.79$
19.   $146\left(\mathrm{~cm}^2\right) \quad$
20.   (a) 16.2 (b) $33.2^{\circ}$ (c) $22.8\left(\mathrm{~m}^2\right) \quad$
21.   $7.31 \mathrm{~cm} \quad$
22.   (a) $D E=7 \mathrm{~cm}$ (b) $B D=2.5 \mathrm{~cm}$ (c) Area $\triangle A B C=39 \mathrm{~cm}^2 \quad$
23.   (a) $A D=8.51$ (b) $B C=6.22$ (c) $\angle A E C=42^{\circ}$ (d) $A B D E=49.9 / 49.8$
24.   (a) show (b) $B C=14 \mathrm{~cm} \quad$ (c) $\angle A B C=38.2^{\circ}$ (d) $\angle P A D=81.8$ (e) $65 \mathrm{~cm}^2\left(64.9 \mathrm{~cm}^2\right)$
25.   $460 cm^2 (459 / 460 / 461)$
26.   (a) show (b) $A G=12.5 \quad$ (c) $\angle G A B=22.4 \quad$ (d) $69.6 \quad$ (e) $59.5 \%$
27.   (a) $B C=8.24$ (b) $A D=13.8$
28.   (a) $BD=6.39$ (b) $\angle BDC = 32.4 ^\circ$ (c) $CD = 6.52$ or 6.53
29.   (a) $E B=9.23 $ (b) $ A B=13.0 $ (c) $ 5.19,5.20 $ (d) $ \angle B E C=32.6,32.7^{\circ} $ (e) $ 140 \mathrm{~cm}^2 \quad$