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MEI Core 2 Trigonometry Topic assessment 1. Find the angle and the length x in the triangle shown below. B 52° 12 cm x cm A 15 cm C [7] 2. Chang walks 5 km on a bearing of 140°, and then walks 3 km on a bearing of 025°. (i) How far is Chang from his starting point? [3] (ii) On what bearing should Chang walk to get back to his starting point? [4] 3. A belt is wrapped around a cylinder of radius 2.5 m as shown. 5m Find the length of the belt. [5] 4. Find the perimeter and area of the shaded sections of these shapes. (i) 3 cm 60° [7] (ii) 8 cm 45° 10 cm [7] 1 of 7 28/10/13 © MEI MEI C2 Trigonometry Assessment solutions 5. Solve these equations for 0 360 (i) cos 0.5 (ii) sin 0.5 (iii) tan 2 [6] 6. Solve these equations for 0 2 . Give your answers as a multiple of . 3 (i) cos 2 (ii) sin 0.5 (iii) tan 3 [6] 7. Solve these equations for 0 2 . Give your answers as a multiple of . 3 (i) cos 2 4 2 (ii) 3tan 1 [6] 8. Solve these equations for 0 360 (i) sin 2 x 12 3 (ii) cos 12 x 0.3 (iii) tan 3x 0.5 [9] 9. Solve these equations for 0 360 (i) cos2 sin 1 (ii) 2sin cos sin 0 (iii) 3 sin cos [4] [4] [3] Total 70 marks 2 of 7 28/10/13 © MEI MEI C2 Trigonometry Assessment solutions Trigonometry Chapter Assessment Solutions sin sin 52 12 15 12 sin 52 sin 15 39.1 or 140.9 The value of cannot be 149.0° as the total of the angles would be greater than 180°. So 39.1 . 1. Using the sine rule: Angle A 180 52 39.08 88.92 x 15 Using the sine rule: sin 88.92 sin 52 15 sin 88.92 x 19.0 cm sin 52 [7] 2. x B O 25° 40° 5 25° 3 40° A (i) Using the cosine rule: x 2 5 2 32 2 3 5 cos 65 x 4.62 km [3] 4.6175 3 5 2 3 4.6175 78.9 Bearing 180 25 78.9 2 (ii) Using the cosine rule: cos 2 2 283.9 [4] 3 of 7 28/10/13 © MEI MEI C2 Trigonometry Assessment solutions 3. cos Adjacent 2.5 1 Hypotenuse 5 2 x 2.5 60 5 Angle of arc with belt on = 360 60 60 240 4 240 240 180 3 4 10 Arc length r 2.5 3 3 tan x 2.5 x 2.5 tan 60 10 2x 3 10 5 tan 60 3 19.1 m(3 s.f.) Total length of belt [5] 4. (i) If the triangle has an angle of 60 at the centre of the circle then it must be an equilateral triangle and so part of the perimeter is 3cm. 60 60 180 3 Arc length r 3 3 Perimeter ( 3) cm 6.14 cm (3 s.f.) 3 3 2 Area Area of sector Area of triangle 3 1 3 3 sin 60 2 2 0.815 cm 2(3 s.f.) Area of sector 21 r 2 0.5 3 2 [7] (ii) Arc length = r Sector area = 21 r 2 45 45 Sector 1: 180 4 Arc length 10 4 4 of 7 5 2 28/10/13 © MEI MEI C2 Trigonometry Assessment solutions Sector area Sector 2: 1 2 Arc length 8 10 2 4 4 25 2 2 Sector area 21 8 2 4 8 Shaded area: Perimeter 2 2 Arc length 1 Arc length 2 5 4 2 2 18.1 cm (3 s.f.) Area Area of sector 1 Area of sector 2 25 8 2 14.1 cm 2(3 s.f.) [7] 5. (i) cos 0.5 Solutions are in the 1st and 4th quadrants. 60 or 360 60 300 60, 300 (ii) sin 0.5 Solutions are in the 3rd and 4th quadrants 180 30 210 or 360 30 330 210, 330 (iii) tan 2 Solutions are in 1st and 3rd quadrants. 63.4 or 180 63.4 243.4 63.4, 243.4 [2] [2] [2] 6. (i) 3 2 Solutions are in 1st and 4th quadrants. 11 or 2 6 6 6 11 , 6 6 cos (ii) sin 0.5 Solutions are in 1st and 2nd quadrants 5 of 7 [2] 28/10/13 © MEI MEI C2 Trigonometry Assessment solutions or 6 5 6 , 6 5 6 6 [2] (iii) tan 3 Solutions are in 1st and 3rd quadrants 4 or 3 3 3 4 , 3 3 7. (i) cos 2 [2] 3 4 3 2 3 has solutions in the 1st and 4th quadrants cos 2 11 or 2 6 6 6 3 has solutions in the 2nd and 4th quadrants cos 2 5 7 or 6 6 6 6 5 7 11 , , , 6 6 6 6 cos (ii) 3 tan 1 1 tan 2 3 1 tan 3 1 tan has solutions in the 1st and 3rd quadrants 3 7 or 6 6 6 1 tan has solutions in the 2nd and 4th quadrants 3 5 11 or 2 6 6 6 6 5 7 11 , , , 6 6 6 6 [3] 2 6 of 7 [3] 28/10/13 © MEI MEI C2 Trigonometry Assessment solutions 8. (i) sin 2 x 21 3 Solutions for 2x are in the 3rd and 4th quadrants. 2 x 180 60 240 or 2 x 360 60 300 or 2 x 360 240 600 or 2 x 360 300 660 x 120, 150, 300, 330 [3] (ii) cos x 0.3 Solutions for 21 x are in the 1st and 4th quadrants, but the one in the 4th quadrant will give a value for x which is out of the range. 1 2 x 72.5 1 2 x 145 (iii) tan 3 x 0.5 Solutions for 3x are in the 1st and 3rd quadrants 3 x 26.6 or 3 x 180 26.6 206.6 or 3 x 360 26.6 386.6 or 3 x 360 206.6 566.6 or 3 x 360 386.6 746.6 or 3 x 360 566.6 926.6 x 8.9, 68.9, 128.9, 188.9, 248.9, 308.9 [3] [3] 9. (i) cos 2 sin 1 1 sin 2 sin 1 sin 2 sin 0 sin sin 1 0 sin 0 sin 1 0 or 0, 180, 360 sin 1 90 0, 90, 180,360 [4] (ii) 2 sin cos sin 0 sin 2 cos 1 0 sin 0 2 cos 1 0 or 0, 180, 360 cos 21 0, 120, 180,240,360 120, 240 [4] (iii) 3 sin cos 3 tan 1 1 tan 3 30,210 [3] 7 of 7 28/10/13 © MEI