# If the solution for $\theta $ of $\cos p\theta +\cos q\theta =0,p>q>0$ are in A.P, then the numerically smallest common difference is A.P.

(a) $\dfrac{\pi }{p+q}$

(b) $\dfrac{2\pi }{p+q}$

(c) $\dfrac{\pi }{2\left( p+q \right)}$

(d) $\dfrac{1}{p+q}$

Last updated date: 19th Mar 2023

•

Total views: 306k

•

Views today: 8.85k

Answer

Verified

306k+ views

Hint: Use the formula of trigonometry $\operatorname{cosx}+cosy=2cos\left( \dfrac{x+y}{2} \right)cos\left( \dfrac{x-y}{2} \right)$. Apply the same to the expression given in the question. Then use the formula $\dfrac{\left( 2n+1 \right)\pi }{2}$ to find values where $\cos \theta =0$.

Complete step-by-step answer:

We are given an equation $\cos p\theta +\cos q\theta =0,p>q>0$.

We need to find its solution, so we will manipulate the given equation to find its solution.

We can use the general formula of trigonometry that is $\operatorname{cosx}+cosy=2cos\left( \dfrac{x+y}{2} \right)cos\left( \dfrac{x-y}{2} \right)$ .

$\cos p\theta +\cos q\theta =0,p>q>0$

Using the formula mentioned above we will get

$2\cos \left( \dfrac{p+q}{2} \right)\theta \cos \left( \dfrac{p-q}{2} \right)\theta =0.....\left( i \right)$

Now to get the solution, equate both its factors one by one with zero.

Now we will equate each factor to zero and we get,

$\cos \left( \dfrac{p+q}{2} \right)\theta =0$

We know the general solution when $\cos \theta =0$ then $\theta $ is equal to $\left( 2n+1 \right)\dfrac{\pi }{2}$ where n belongs to an integer.

So it’s clear from the general solution that,

$\left( \dfrac{p+q}{2} \right)\theta =\left( 2n+1 \right)\dfrac{\pi }{2}....\left( ii \right)$

where n belongs to integers.

And, similarly again we use the general solution formula, $\cos \theta =0$ then $\theta $ is equal to $\left( 2n+1 \right)\dfrac{\pi }{2}$ where n belongs to an integer for the second factor as well.

$\cos \left( \dfrac{p-q}{2} \right)\theta =0$

So,

\[\left( \dfrac{p-q}{2} \right)\theta =\left( 2n+1 \right)\dfrac{\pi }{2}....\left( iii \right)\]

where n belongs to integer

Now it is given in the question that solution of it would form an A.P so observing equation (ii) and equation (iii) we can observe that on putting the value of $n=1,2,3,,,,$ and so on we will get a common difference in each term i.e.

For equation (ii)

When $n=1$

$\left( \dfrac{p+q}{2} \right)\theta =\dfrac{3\pi }{2}$

When $n=2$

$\left( \dfrac{p+q}{2} \right)\theta =\dfrac{5\pi }{2}$

And so on.

We get a common difference $\pi $ for every successive value of n.

So,

$\begin{align}

& \left( \dfrac{p+q}{2} \right)\theta =\pi \\

& \theta =\dfrac{2\pi }{p+q} \\

\end{align}$

We get one of the solutions of the given equation.

$\theta =\dfrac{2\pi }{p+q}.....\left( iv \right)$

Now for equation (iii)

When $n=1$

$\left( \dfrac{p-q}{2} \right)\theta =\dfrac{3\pi }{2}$

When $n=2$

$\left( \dfrac{p-q}{2} \right)\theta =\dfrac{5\pi }{2}$

And so on.

We get a common difference $\pi $ for every successive value of n.

So,

$\begin{align}

& \left( \dfrac{p-q}{2} \right)\theta =\pi \\

& \theta =\dfrac{2\pi }{p-q} \\

\end{align}$

We get one of the solutions of the given equation.

$\theta =\dfrac{2\pi }{p-q}....\left( iv \right)$

Now we need to find the smallest solution for $\theta $,

So,

We will observe equation (iii) and (iv) and it is given in the question that $p>q>0$, so $p-q$ is less than $p+q$ and we know that if denominator value is greater, then fraction will be smaller. So, $\theta =\dfrac{2\pi }{p+q}$ the value of $\theta $ is much smaller than $\theta =\dfrac{2\pi }{p-q}$.

So the correct option is b.

Note: The possibility of mistake that could be done here is not considering both solutions of equation i.e. forgetting to evaluate the solution $\theta =\dfrac{2\pi }{p-q}$ although it will be rejected later on as it is not the smallest solution for $\theta $ . But we have to show that process in the solution.

Complete step-by-step answer:

We are given an equation $\cos p\theta +\cos q\theta =0,p>q>0$.

We need to find its solution, so we will manipulate the given equation to find its solution.

We can use the general formula of trigonometry that is $\operatorname{cosx}+cosy=2cos\left( \dfrac{x+y}{2} \right)cos\left( \dfrac{x-y}{2} \right)$ .

$\cos p\theta +\cos q\theta =0,p>q>0$

Using the formula mentioned above we will get

$2\cos \left( \dfrac{p+q}{2} \right)\theta \cos \left( \dfrac{p-q}{2} \right)\theta =0.....\left( i \right)$

Now to get the solution, equate both its factors one by one with zero.

Now we will equate each factor to zero and we get,

$\cos \left( \dfrac{p+q}{2} \right)\theta =0$

We know the general solution when $\cos \theta =0$ then $\theta $ is equal to $\left( 2n+1 \right)\dfrac{\pi }{2}$ where n belongs to an integer.

So it’s clear from the general solution that,

$\left( \dfrac{p+q}{2} \right)\theta =\left( 2n+1 \right)\dfrac{\pi }{2}....\left( ii \right)$

where n belongs to integers.

And, similarly again we use the general solution formula, $\cos \theta =0$ then $\theta $ is equal to $\left( 2n+1 \right)\dfrac{\pi }{2}$ where n belongs to an integer for the second factor as well.

$\cos \left( \dfrac{p-q}{2} \right)\theta =0$

So,

\[\left( \dfrac{p-q}{2} \right)\theta =\left( 2n+1 \right)\dfrac{\pi }{2}....\left( iii \right)\]

where n belongs to integer

Now it is given in the question that solution of it would form an A.P so observing equation (ii) and equation (iii) we can observe that on putting the value of $n=1,2,3,,,,$ and so on we will get a common difference in each term i.e.

For equation (ii)

When $n=1$

$\left( \dfrac{p+q}{2} \right)\theta =\dfrac{3\pi }{2}$

When $n=2$

$\left( \dfrac{p+q}{2} \right)\theta =\dfrac{5\pi }{2}$

And so on.

We get a common difference $\pi $ for every successive value of n.

So,

$\begin{align}

& \left( \dfrac{p+q}{2} \right)\theta =\pi \\

& \theta =\dfrac{2\pi }{p+q} \\

\end{align}$

We get one of the solutions of the given equation.

$\theta =\dfrac{2\pi }{p+q}.....\left( iv \right)$

Now for equation (iii)

When $n=1$

$\left( \dfrac{p-q}{2} \right)\theta =\dfrac{3\pi }{2}$

When $n=2$

$\left( \dfrac{p-q}{2} \right)\theta =\dfrac{5\pi }{2}$

And so on.

We get a common difference $\pi $ for every successive value of n.

So,

$\begin{align}

& \left( \dfrac{p-q}{2} \right)\theta =\pi \\

& \theta =\dfrac{2\pi }{p-q} \\

\end{align}$

We get one of the solutions of the given equation.

$\theta =\dfrac{2\pi }{p-q}....\left( iv \right)$

Now we need to find the smallest solution for $\theta $,

So,

We will observe equation (iii) and (iv) and it is given in the question that $p>q>0$, so $p-q$ is less than $p+q$ and we know that if denominator value is greater, then fraction will be smaller. So, $\theta =\dfrac{2\pi }{p+q}$ the value of $\theta $ is much smaller than $\theta =\dfrac{2\pi }{p-q}$.

So the correct option is b.

Note: The possibility of mistake that could be done here is not considering both solutions of equation i.e. forgetting to evaluate the solution $\theta =\dfrac{2\pi }{p-q}$ although it will be rejected later on as it is not the smallest solution for $\theta $ . But we have to show that process in the solution.

Recently Updated Pages

If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

KCN reacts readily to give a cyanide with A Ethyl alcohol class 12 chemistry JEE_Main

Trending doubts

What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Tropic of Cancer passes through how many states? Name them.

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

Name the Largest and the Smallest Cell in the Human Body ?