Question

The general value of \[\theta \] in the equation \[2\sqrt 3 \cos \theta = \tan \theta \], is
A. \[2n\pi \pm \dfrac{\pi }{6}\]
B. \[2n\pi \pm \dfrac{\pi }{4}\]
C. \[n\pi + {( - 1)^n}\dfrac{\pi }{3}\]
D. \[n\pi + {( - 1)^n}\dfrac{\pi }{4}\]

Answer 1

Hint: To solve this question, we will use the trigonometric formulas of \[\cos \theta \] and \[\tan \theta \]. The angles of the triangle are \[\pi \]. When solving trigonometry problems, the values of the trigonometric functions for \[{0^ \circ }\], \[{30^ \circ }\], \[{45^ \circ }\], \[{60^ \circ }\], and \[{90^ \circ }\] are frequently applied. The sides of the right triangle, such as the neighbouring side, opposite side, and hypotenuse side, are used to determine each of these trigonometric ratios.

Formula Used: The trigonometric formulas of tangent are:
\[2\sqrt 3 \cos \theta = \tan \theta \]
\[ \Rightarrow 2\sqrt 3 \cos \theta = \dfrac{{\sin \theta }}{{\cos \theta }}\]
\[ \Rightarrow 2\sqrt 3 {\cos ^2}\theta = \sin \theta \]

Complete step by step solution: We have, \[2\sqrt 3 \cos \theta = \tan \theta \]
\[ \Rightarrow 2\sqrt 3 \cos \theta = \dfrac{{\sin \theta }}{{\cos \theta }}\]
\[ \Rightarrow 2\sqrt 3 {\cos ^2}\theta = \sin \theta \]
\[ \Rightarrow 2\sqrt 3 {\sin ^2}\theta + \sin \theta - 2\sqrt 3 = 0\]
\[ \Rightarrow \sin \theta = \dfrac{{ - 1 + 7}}{{4\sqrt 3 }}\]
\[ \Rightarrow \sin \dfrac{{ - 8}}{{4\sqrt 3 }}\] (Impossible)
\[ \Rightarrow \sin \theta = \dfrac{6}{{4\sqrt 3 }} = \dfrac{{\sqrt 3 }}{2}\]
Therefore, \[\theta = n\pi + {( - 1)^n}\dfrac{\pi }{3}\]
The basic trigonometry formula \[\cos \theta = \dfrac{1}{{\tan \theta }}\sin \theta = \dfrac{1}{{\cos \theta }}\cos \theta = \dfrac{1}{{\sin \theta }}\tan \theta = \dfrac{1}{{\cos \theta }}\]. The equations \[\cos \theta = \dfrac{1}{{\sqrt 2 }}\] and \[\tan \theta = 1\] gives the general value of theta. The value matching the equations \[\sin = \sin \] and \[\cos = \cos \] is most generally expressed as\[n{\rm{ }} = {\rm{ }} + \].
Periodic Identities: These are trigonometry formulas that assist in determining the values of trig functions for an angle shift of \[/2\],etc.
Formulas for trigonometry can be applied to a wide range of problems. These issues could involve Pythagorean identities, product identities, trigonometric ratios (sin, cos, tan, sec, cosec, and cot), etc.
The nature of all trigonometric identities is cyclicity. After this periodicity is constant, they repeat. Varies depending on the trigonometric identity.

Option ‘C’ is correct
Note: The reciprocals of the fundamental trigonometric ratio’s sine, cosine, and tangent are cosecant, secant, and cotangent. A right-angled triangle is used as a model to determine each reciprocal identity. Right-angle triangle lengths and angles are measured using trigonometry values of various ratios, including sine, cosine, tangent, secant, cotangent, and cosecant. When solving trigonometry problems, the values of the trigonometric functions for\[{0^ \circ }\],\[{30^ \circ }\],\[{45^ \circ }\],\[{60^ \circ }\], \[{90^ \circ }\]and are frequently employed.