Question 14:
(a) Solve the equation 3 sin 2x = 4 cosx for 0∘ ≤ x ≤ 360∘. [4 marks]
(b) Solve the simultaneous equations 2 cos (x – y) = √3 and 2 cos (x + y) = 1, where x and y are both acute angles. [4 marks]
Solution:
(a)
3sin2x=4cosx3(2sinxcosx)=4cosx6sinxcosx=4cosx6sinxcosx−4cosx=02cosx(3sinx−2)=02cosx=0cosx=0x=90o, 270o3sinx−2=03sinx=2sinx=23x=sin−123x=41.81o, (180o−41.81o)x=41.81o, 138.19o∴x=41.81o, 90o, 138.19o, 270o
(b)
2cos(x−y)=√3cos(x−y)=√32(x−y)=cos−1(√32)(x−y)=30 ….. (1)2cos(x+y)=1cos(x+y)=12(x+y)=cos−1(12)(x+y)=60 ….. (2)x−y=30 ….. (1)x+y=60 ….. (2)(1)+(2),2x=90x=45From (1),45−y=30y=15
(a) Solve the equation 3 sin 2x = 4 cosx for 0∘ ≤ x ≤ 360∘. [4 marks]
(b) Solve the simultaneous equations 2 cos (x – y) = √3 and 2 cos (x + y) = 1, where x and y are both acute angles. [4 marks]
Solution:
(a)
3sin2x=4cosx3(2sinxcosx)=4cosx6sinxcosx=4cosx6sinxcosx−4cosx=02cosx(3sinx−2)=02cosx=0cosx=0x=90o, 270o3sinx−2=03sinx=2sinx=23x=sin−123x=41.81o, (180o−41.81o)x=41.81o, 138.19o∴x=41.81o, 90o, 138.19o, 270o
(b)
2cos(x−y)=√3cos(x−y)=√32(x−y)=cos−1(√32)(x−y)=30 ….. (1)2cos(x+y)=1cos(x+y)=12(x+y)=cos−1(12)(x+y)=60 ….. (2)x−y=30 ….. (1)x+y=60 ….. (2)(1)+(2),2x=90x=45From (1),45−y=30y=15