Question

The smallest positive angle which satisfies the equation \[2{\sin ^2}\theta + \sqrt 3 \cos \theta + 1 = 0\]
A. \[\dfrac{{5\pi }}{6}\]
В. \[\dfrac{{2\pi }}{3}\]
C. \[\dfrac{\pi }{3}\]
D. \[\dfrac{\pi }{6}\]

Answer 1

Hint: A trigonometric identity is an equation based on trigonometry which is always true. The sum and differences can be determined using \[sin(\alpha + \beta ) = sin\alpha cos\beta + cos\alpha sin\beta \]and\[sin(\alpha - \beta ) = sin\alpha cos\beta - cos\alpha sin\beta \]. In this case, the smallest positive angle which satisfies the equation \[2{\sin ^2}\theta + \sqrt 3 \cos \theta + 1 = 0\]can be obtained by solving them using trigonometry identities and sum and differences of two angles concept. To simplify the equation in this question, we will utilize a trigonometric formula, input the values, and answer the problem. But to do that, we must first make it smaller.





Complete step by step solution:We have given the equation, according to the question:
\[2{\sin ^2}x + \sqrt 3 \cos x + 1 = 0\]
Expand the given equation using trig identities:
\[ \Rightarrow 2\left( {1 - {{\cos }^2}\theta } \right) + \sqrt 3 \cos \theta + 1 = 0\]
Multiply\[2\]with the terms inside the parentheses:
\[ \Rightarrow 2 - 2{\cos ^2}\theta + \sqrt 3 \cos \theta + 1 = 0\]
Simplify the above resultant equation:
\[ \Rightarrow 2{\cos ^2}\theta - \sqrt 3 \cos \theta - 3 = 0\]
Now, we have to expand the equation obtained:
\[ \Rightarrow 2{\cos ^2}\theta - 2\sqrt 3 \cos \theta + \sqrt 3 \cos \theta - 3 = 0\]
Expand them in terms of cos:
\[ \Rightarrow 2\cos \theta (\cos \theta - \sqrt 3 ) + \sqrt 3 (\cos \theta - \sqrt 3 ) = 0\]
Now solve the above equation in much less complicated form:
\[ \Rightarrow (2\cos \theta + \sqrt 3 )(\cos \theta - \sqrt 3 ) = 0\]
In order to find the value of cos, multiply two terms:
\[ \Rightarrow 2\cos \theta + \sqrt 3 = 0\]
Thus the equation can be written as:
\[\cos \theta - \sqrt 3 = 0\]
Now, solve for\[\cos \theta \]:
\[\therefore \cos \theta = - \dfrac{{\sqrt 3 }}{2}\]or\[\cos \theta = \sqrt 3 \]is not possible.
This can also be written as,
\[ \Rightarrow x = 2n\pi \pm \dfrac{{5\pi }}{6},n \in Z\]
For\[n = 0\],
The value of\[x\] is\[ \pm \dfrac{{5\pi }}{6}\]
Hence, the smallest positive angle which satisfies the equation\[2{\sin ^2}\theta + \sqrt 3 \cos \theta + 1 = 0\]is\[\dfrac{{5\pi }}{6}\]



Option ‘A’ is correct

Note: We must commit the key formula to memory in order to solve this kind of problem, and we can only do this through practice. Therefore, it is important to understand how to use trigonometric identities and to build identities in the correct order. It simplifies the problem and aids in your quest for the right response. This method makes it very simple for us to solve this kind of issue.