3 a Determine the exact value of sin tan13 cos1 tan V3cos3 6

3) (a) Determine the exact value of sin tan-13- cos1 tan- V3-cos-3 [6 marks] (b) Find OP 360° such tht cos 2x = cos z. [6 marks

Solution

3 a) We have given sin[arctan(sqrt(3))-arccos(1/3)]

sin[arctan(sqrt(3))-arccos(1/3)]

Let x=arctan(sqrt(3))

tan(x)=sqrt(3)

x=pi/3

sin[arctan(sqrt(3))-arccos(1/3)]=sin[x-arccos(1/3)]

=sin[pi/3-arccos(1/3)]

=sin(pi/3)cos(arccos(1/3))-cos(pi/3)sin(arccos(1/3)) since sin(A-B)=sinAcosB-cosAsinB

using sin(arccos(x))=sqrt(1-x^2)

sin(arccos(1/3))=sqrt(1-1/9)=sqrt(8)/3

=(sqrt(3)/2)*(1/3)-(1/2)*(sqrt(8)/3)

=[sqrt(3)-sqrt(8)]/6

sin[arctan(sqrt(3))-arccos(1/3)]=[sqrt(3)-sqrt(8)]/6

b) We have given cos(2x)=cosx,0<=x<=360⁰

cos(2x)-cosx=0

plug cos(2x)=2cos²x-1

2cos²x-1-cosx=0

2cos²x-cosx-1=0

2cos²x-2cosx+cosx-1=0

2cosx(cosx-1)+1(cosx-1)=0

(cosx-1)(2cosx+1)=0

cosx-1=0 ,2cosx+1=0

cosx=1,cosx=-1/2

x=0,2pi when cosx=1,x=2pi/3,4pi/3 when cosx=-1/2

x=0,2pi,2pi/3,4pi/3