Solve the equation cos3t=cos2t?

3 Answers
May 22, 2017

"The Soln. Set is "{2/5kpi : k in ZZ}.

Explanation:

cos2theta=cos3theta.

rArr cos3theta-cos2theta=0.

rArr -2sin((3theta+2theta)/2)*sin((3theta-2theta)/2)=0.

rArr -2sin((5theta)/2)sin(theta/2)=0.

rArr sin((5theta)/2)=0, or, sin(theta/2)=0.

Knowing that, sinx=o rArr x=kpi, k in ZZ, we have,

5/2theta=kpi, or, theta/2=kpi, k in ZZ.

theta=2/5kpi, or, theta=2kpi, k in ZZ.

rArr theta in {2/5kpi}uu{2kpi}, k in ZZ.

But, we observe that, {2kpi : k in ZZ} sub {2/5kpi : k in ZZ}, hence

"The Soln. Set is "{2/5kpi : k in ZZ}.

Enjoy Maths.!

May 22, 2017

Solution is theta=(2npi)/5, where n is an integer.

Explanation:

We will use here cosB-cosA=2sin((A+B)/2)sin((A-B)/2)

Hence, we ccan write cos2theta=cos3theta as

cos2theta-cos3theta=0

or 2sin((3theta+2theta)/2)sin((3theta-2theta)/2)=0

or 2sin((5theta)/2)sin(theta/2)=0

i.e. either (5theta)/2=npi or theta/2=npi, where n is an integer.

i.e. either theta=(2npi)/5 or theta=2npi, where n is an integer.

Observe that the latter set (i.e. where theta=2npi) is a subset of former set (i.e. where theta=(2npi)/5). This is so when n is a multiple of 5.

Hence solution is theta=(2npi)/5, where n is an integer.

May 22, 2017

t = 2kpi
t = (2kpi)/5

Explanation:

cos 3t = cos 2t
3t = +- 2t

a. 3t = 2t
t = 0 or t = 2kpi

b. 3t = - 2t
5t = 0 or 5t = 2kpi
t = (2kpi)/5
Check by calculator, with k = 1
t = (2pi)/5 = 72^@ --> cos 2t = cos 144^@ = - 0.809 -->
cos 3t = cos 216^@ = - 0.809. OK
t = 2pi --> cos 2t = cos 4pi = 1 and cos 3t = cos 6pi = 1. OK