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If tan mƟ = tan nƟ, then show that the value Ɵ form an A.P. Also find the common difference of the A.P?

If tan mƟ = tan nƟ, then show that the value Ɵ form an A.P. Also find the common difference of the A.P?

Grade:11

2 Answers

Arun
25758 Points
5 years ago
By general solution,mƟ = kπ + nƟ(m-n)Ɵ = kπƟ = kπ/(m-n)Hence we see that value of Ɵ makes an A.P. with common difference of π/(m-n)Because k is an integerHence terms are .....π/(m-n), 2π/(m-n), 3π/(m-n), .......
Vikas TU
14149 Points
5 years ago
Dear Student,
tan(m theta) = tan(n theta)
Take the inverse tangent of both sides:
m theta = N theta + pi N  for  N belongs to Z
Subtract b theta from both sides:
theta (m-n) = pi N  for  N belongs to Z
Divide both sides by m-n:
theta = (pi N)/(m-n)  for  N belongs to Z
so, putting value of N=1, theta=pi/(m-n)
putting value of N=2, theta=2pi/(m-n) and so on..
so the values of theta form an ap
and common difference = (N+1)pi/(m-n)-N pi/(m-n)=pi/(m-n)
Therefore,proved that values of theta form an AP with common difference pi/(m-n).
Cheers!!
Regards,
Vikas (B. Tech. 4th year
Thapar University)

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