Answer

Verified

414.3k+ views

**Hint:**This is the basic question of trigonometry so we have to find in which coordinate these values exist. And we use a simple conversion of $\cos ec\theta $ into terms of $\sin \theta $ . This conversion is we do just for our benefit purpose. And have to find the general solution for each equation and after that we take the intersection of them. And write the result in the form of a general equation by defining $n$.

**Complete step-by-step answer:**

In this we have to remember the basic coordinate system of all the trigonometric function and their basic general solution formula

Here we choose first equation $\tan \theta = \sqrt 3 $

And solution for $\theta = \dfrac{\pi }{3},\dfrac{{4\pi }}{3},\dfrac{{7\pi }}{3}.....$

Now we take second equation $\cos ec\theta = - \dfrac{2}{{\sqrt 3 }}$

So we convert this in term of $\sin \theta $

By using formula $\cos ec\theta = \dfrac{1}{{\sin \theta }}$

So we have $\dfrac{1}{{\sin \theta }} = - \dfrac{2}{{\sqrt 3 }}$

And $\sin \theta = - \dfrac{{\sqrt 3 }}{2}$

Now we know that $\sin \theta $ is negative in $IIIrd$ and $ivth$ coordinate

So $\theta = \pi + \dfrac{\pi }{3},2\pi - \dfrac{\pi }{3}$ this value is repeated after every $n\pi $ interval.

$\theta = \dfrac{{4\pi }}{3},\dfrac{{5\pi }}{3}$

Now the common value in both the equation is $\dfrac{{4\pi }}{3}$

So the general solution is $n\pi + \dfrac{{4\pi }}{3}:n \in I$

**Option A is the correct answer.**

**Note:**We have to choose only that value of $\theta $ which is common because this value is repeated after every $n\pi $ interval. And we can find this by using graphical methods .

For graphical method have follow this steps

* Make graph of each function

* Denote all the values and compare where both take the same value.

Recently Updated Pages

Cryolite and fluorspar are mixed with Al2O3 during class 11 chemistry CBSE

Select the smallest atom A F B Cl C Br D I class 11 chemistry CBSE

The best reagent to convert pent 3 en 2 ol and pent class 11 chemistry CBSE

Reverse process of sublimation is aFusion bCondensation class 11 chemistry CBSE

The best and latest technique for isolation purification class 11 chemistry CBSE

Hydrochloric acid is a Strong acid b Weak acid c Strong class 11 chemistry CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

The Buddhist universities of Nalanda and Vikramshila class 7 social science CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Name 10 Living and Non living things class 9 biology CBSE

Which are the Top 10 Largest Countries of the World?

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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

Who founded the Nalanda University 1 Mauryan 2 Guptas class 6 social science CBSE