Question

The general solution of \[{\sin ^2}\theta \sec \theta + \sqrt 3 \tan \theta =
       0\] is
A. \[\theta = n\pi + {( - 1)^{n + 1}}\dfrac{\pi }{3},\theta = n\pi ,n \in z\]
B. \[\theta = {n^\pi }\], \[n \in z\]
C. \[\theta = n\pi + {( - )^{n + 1}}\dfrac{\pi }{3},n \in z\]
D. \[\theta = \dfrac{{n\pi }}{2},n \in Z\]

Answer 1

Hint: In order to answer this question, we will use the reciprocal relationships, by which \[\sec \theta \]will be equal to\[\dfrac{1}{{\cos \theta }}\], as well as the quotient relationship\[\tan \theta \], which is equal to\[\dfrac{{\sin \theta }}{{\cos \theta }}\]. By substituting these values into the given equation, we obtain an equation from which the common term \[\sin \theta \] is taken, and by moving all the other terms to the opposite side. As a result, the equation can have a general solution.

Formula Used: The trigonometric formulas of sin are:
\[{\sin ^2}\theta \]\[ = \dfrac{1}{{\cos \theta }}\]
\[\tan \theta \]\[ = \dfrac{{\sin \theta }}{{\cos \theta }}\]

Complete step by step solution: We are given an equation with \[{\sin ^2}\theta \] and \[\tan \theta \], we have to substitute the values according to the quotient relation and reciprocal relation,
We will get the following equation:
\[{\sin ^2}\theta \sec \theta + \sqrt 3 \tan \theta = 0\]
Substitute \[\sec \theta \] is equal to\[\dfrac{1}{{\cos \theta }}\], in accordance with quotient relation, we will get,
\[{\sin ^2}\theta \dfrac{1}{{\cos \theta }} + \sqrt 3 \tan \theta = 0\]
Substitute \[\tan \theta \] is equal to \[\dfrac{{\sin \theta }}{{\cos \theta }}\], in accordance with reciprocal relation, we will get,
\[{\sin ^2}\theta \dfrac{1}{{\cos \theta }} + \sqrt 3 \dfrac{{\sin \theta }}{{\cos \theta }} = 0\]
Taking \[\dfrac{1}{{\cos \theta }}\]to the Right-hand side we will get,
\[\sin \theta \](\[\sin \theta \]+\[\sqrt 3 \])\[ = 0\]
Since, \[\sin \theta \] is a common term, we take 1 term from the above equation,
\[\sin \theta + \sqrt 3 = 0\]
Taking \[\sqrt 3 \]to the left side, positive \[\sqrt 3 \]will become negative \[\sqrt 3 \], as follows:
\[\sin \theta = - \sqrt 3 \]
\[\theta = n\pi \]

 Option ‘B’ is correct

Note: To get the general solution for the given equation, \[{\sin ^2}\theta \sec \theta + \sqrt 3 \tan \theta = 0\]we have to use trigonometry identities that are equal to their reciprocal relations and to their quotient relations in accordance with the required solution. The angle \[\theta \] can be generated from the given equation by substituting it in the given equation. In order to find the solution for the given trigonometrical identity we should start solving from the left-hand side or right-hand side and by applying trigonometrical relations we should come to the other side.