
If \[\tan n\theta = \tan m\theta \] , then find the different value of \[\theta \] will be in which progression.
A. A.P
B. G.P
C. H.P
D. None of these
Answer
163.2k+ views
Hints First obtain the general solution for the given equation, then substitute 1,2,3,… for n in the obtained general solution to observe the progression. Subtract the second term from the first term and then the third term from the second term to observe whether the difference is the same or not. If the difference is the same then the progression is in A.P.
Formula used
The general solution of \[\tan \theta = \tan \beta \] is \[\theta = n\pi + \beta ,n = 1,2,3,....\] .
Complete step by step solution
The given equation is \[\tan n\theta = \tan m\theta \].
Therefore, the general solution is,
\[n\theta = N\pi + m\theta \]
\[n\theta - m\theta = N\pi \]
\[\theta = \dfrac{{N\pi }}{{n - m}}\]
Now, substitute N=1,2,3,… for evaluation.
The sequence will be \[\left\{ {\dfrac{\pi }{{n - m}},\dfrac{{2\pi }}{{n - m}},\dfrac{{3\pi }}{{n - m}},...} \right\}\] .
Therefore, the common difference of second term and first term is \[\dfrac{{2\pi }}{{n - m}} - \dfrac{\pi }{{n - m}} = \dfrac{\pi }{{n - m}}\]
And third term and second term is \[\dfrac{{3\pi }}{{n - m}} - \dfrac{{2\pi }}{{n - m}} = \dfrac{\pi }{{n - m}}\], hence the sequence is in A.P.
The correct option is A.
Note Sometimes students write the general solution as \[n\theta = m\theta \] , hence they get \[\theta = 0\] as the answer but this is not correct. Use the general formula for \[\tan \theta = \tan \beta \] as \[\theta = n\pi + \beta ,n = 1,2,3,....\] and calculate to obtain the required answer.
Formula used
The general solution of \[\tan \theta = \tan \beta \] is \[\theta = n\pi + \beta ,n = 1,2,3,....\] .
Complete step by step solution
The given equation is \[\tan n\theta = \tan m\theta \].
Therefore, the general solution is,
\[n\theta = N\pi + m\theta \]
\[n\theta - m\theta = N\pi \]
\[\theta = \dfrac{{N\pi }}{{n - m}}\]
Now, substitute N=1,2,3,… for evaluation.
The sequence will be \[\left\{ {\dfrac{\pi }{{n - m}},\dfrac{{2\pi }}{{n - m}},\dfrac{{3\pi }}{{n - m}},...} \right\}\] .
Therefore, the common difference of second term and first term is \[\dfrac{{2\pi }}{{n - m}} - \dfrac{\pi }{{n - m}} = \dfrac{\pi }{{n - m}}\]
And third term and second term is \[\dfrac{{3\pi }}{{n - m}} - \dfrac{{2\pi }}{{n - m}} = \dfrac{\pi }{{n - m}}\], hence the sequence is in A.P.
The correct option is A.
Note Sometimes students write the general solution as \[n\theta = m\theta \] , hence they get \[\theta = 0\] as the answer but this is not correct. Use the general formula for \[\tan \theta = \tan \beta \] as \[\theta = n\pi + \beta ,n = 1,2,3,....\] and calculate to obtain the required answer.
Recently Updated Pages
Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Instantaneous Velocity - Formula based Examples for JEE

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations

NCERT Solutions for Class 11 Maths In Hindi Chapter 1 Sets

JEE Advanced 2025 Notes
