Question

The roots of the equation\[\]\[1 - cos\theta = sin\theta \]. \[sin\dfrac{\theta }{2}\]is
A. \[k\pi ,k \in I\]
B. \[2k\pi ,k \in I\]
C. \[k\pi 2,k \in I\]
D. none of these

Answer 1

Hint: To solve this question, we will use the formula from trigonometry \[\cos \theta \]and \[\sin 2\theta \]. we will derive the value of \[k\]and the \[\theta \].also we will simplify the equation using trigonometry formulas and get the resultant values. The relationship between the length of the sides of the right triangle and the measurement of the angles is known as the trigonometric ratio. Formulas involving trigonometric functions are known as trigonometric idents. For each possible value of the variables, these identities hold true.

Formula Used: The trigonometric formula of cos and sin are
\[1 - \cos \dfrac{\theta }{2}\]
\[\sin \theta = 2\sin \dfrac{\theta }{2}.\cos \dfrac{\theta }{2}\]

Complete step by step solution: The roots of the equation\[\]\[1 - cos\theta = sin\theta \]. \[sin\dfrac{\theta }{2}\]
By using the trigonometry formula
\[1 - cos\theta = sin\theta \]. \[sin\dfrac{\theta }{2}\]
\[2{\sin ^2}\dfrac{\theta }{2} = \]\[2\sin \dfrac{\theta }{2}.\cos \dfrac{\theta }{2}.\sin \dfrac{\theta }{2}\]
Simplify 2 on both sides
\[{\sin ^2}\theta = {\sin ^2}\dfrac{\theta }{2}.\cos \dfrac{\theta }{2}\]
\[{\sin ^2}\theta - {\sin ^2}\dfrac{\theta }{2}.\cos \dfrac{\theta }{2} = 0\]
Take \[\sin \theta \] common
\[{\sin ^2}\dfrac{\theta }{2}(1 - \cos \dfrac{\theta }{2}) = 0\]
Using trigonometry formula
\[1 - \cos \theta = 2{\sin ^2}\dfrac{\theta }{2}\]
Divide by \[\dfrac{\theta }{2}\]
\[1 - \cos \dfrac{\theta }{2} = 2{\sin ^2}\dfrac{\theta }{4}\]
Using this formula in the equation
\[{\sin ^2}\dfrac{\theta }{2}(1 - \cos \dfrac{\theta }{2}) = 0\]
\[{\sin ^2}\dfrac{\theta }{2}.2{\sin ^2}\dfrac{\theta }{4} = 0\]
Take \[2\] to the other side
\[{\sin ^2}\dfrac{\theta }{2}.{\sin ^2}\dfrac{\theta }{4} = 0\]
\[{\sin ^2}\dfrac{\theta }{2} = 0\] or \[{\sin ^2}\dfrac{\theta }{4} = 0\]
Taking square root on both sides
\[\sin \dfrac{\theta }{2} = 0\] or \[\sin \dfrac{\theta }{4} = 0\]
In trigonometry, if \[\sin \theta = 0\] then \[\theta = k\pi \]where \[k \in i\]
Then \[\dfrac{\theta }{2} = k\pi \] or \[\dfrac{\theta }{4} = k\pi \] where \[k \in i\]
\[\theta = 2k\pi \] or \[\theta = 4k\pi \]where \[k \in i\]

Option ‘B’ is correct

Note: The trigonometry formula should be used correctly, and we have to modify the formula according to our need then the formula should vary. If we take square root on both sides, then it should be noted carefully. Three fundamental trigonometric operations are Sin, Cos, and Tan. The problem utilises trigonometric functions like sine, cosine, tangent, cotangent, cosecant, and secant. Trigonometric formulas are utilised to evaluate the problem. We can resolve issues with the angles and sides of a right triangle by using the various trigonometric identities.