Activity 1
In triangle ABC, ^B = 37° and AC = 16 cm. ^C = 90°. Calculate the length of AB and BC (correct to one decimal place). (3)
Solution To calculate the length of AB, use 37° as the reference angle, then AC = 16 cm is the opposite side and AB is the hypotenuse. Use the sine ratio. sin 37° = opp =16 hyp AB AB sin 37° = 16 AB = 16 = 26,6 cm sin37° To find the length of BC, you can use cos 37° = adj = BC/26.6 hyp 26,6 cos 37° = BC BC = 21,2 cm (to one decimal place) You can also use Pythagoras’ theorem: AB2 = AC2 + BC2 [3] [3] |
Activity 2
In triangle PQR, PQ = 12,3 m and PR = 13 m. Calculate the size of ^Q . (2)
[2]
Solution Use PQ and PR. tan θ = opp = 13 12.3 θ = tan–1 (13/12.3) = 46,58° [2] |
Activity 3
Solve ∆XYZ in which z = 7,3 m, ^X = 43° and ^Y = 96°. Give your solutions correct to 3 decimal places. (4)
[4]
Solution The angle opposite the known side is not given, but you can work it out. ^Z = 180° – (43° + 96°) (sum angles of ∆) ^Z = 41° To find y: y = 7,3 sin 96° sin41° y = 7,3 sin 96° sin41° y = 11,066 m Using the sine rule again to find x: x = 7,3 sin43° sin41° x = 7,3 sin 43° sin41° x = 7,589 m [4] |
Activity 4
1. PQRS is a trapezium with PQ // SR, PQ = PS, SR = 10 cm,
QR = 7 cm, ^R = 63°.
Calculate:
Solutions
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When solving triangles, start with the triangle which has most information (i.e. triangle with three sides or two sides and an angle or two angles and a side given)
Activity 5
In the diagram alongside, AC = 7 cm,
DC = 3 cm, AB = AD, DCA = 60°,
DAB = β and ABD = θ.
Show that BD = √37 sin β
sinθ
[3]
Solution AD2 = AC2 + CD2 − 2AC.CD cos 60 ° = (7)2 + (3)2 – 2 × 7 × 3 × 0,5 AD2 =58 – 21 AD2 = 37 AD = √37 P Applying sine rule: BD = AD ⇒ BD = AD sin β but AD = √37 sinβ sinθ sinθ ∴BD = √37 sin β sinθ [3] |
Activity 6
Solutions 1.1 sin 2x = AB ∴ AB = r sin 2x (2) r 1.2 AKC = 90° + x [ext. angle of ∆CBK] (2) 1.3 AK = r ∴ AK = r sinx sin x sin(90° + x) cos x r sin x AK = cos x = r sin x = 1 = 2 AB r sin 2x r cos x.2 cos x sin x 2 cos2x 3 ∴ cos2x = 3/4 cos x = √3/2 Hence x = 30º (7) 2.1 In ∆HLB, tan 40° = 3 LB [∆HLB is right-angled, so use a trig ratio] LB = 3 tan40º LB = 3,5752… ≈ 3,58 metres (3) 2.2 In ∆ABL, [ΔABL not right-angled. You have two sides and included angle, so use the Cosine Rule] AB2 = AL2 + BL2 – 2(AL)(BL).cos L AB2 = (5,2)2 + (3,58)2 – 2(5,2)(3,58).cos 113° AB2 = 54,40410… m2 AB = 7,38 m (3) 2.3 Area ∆ABL = ½ AL × BL × sin ALB = ½ (5,2) × (3,58) × sin 113° = 8,56805… ≈ 8,57 m2 (3) 3.1 cos 64,750° = CM ∴ AC = CM = 50m = 117,21 (2) AC cos64.75° 0.426569 3.2 tanPAC = PC AC PAC = tan−1 (32/AC) = 15,27° (2) 3.3 DC2 = AC2 + AD2 − 2AC.ADcos(90° − 64,75°) DC2 = (117,21 )2 + (40)2 − 2(117,21).40cos(25,25°) = 6857,289 DC = 82,81m (4) [28] |