Question

If $\sin \theta +\cos \theta =1$ then the value of \[\sin 2\theta \] is equal to:
A. 1
B. \[\dfrac{1}{2}\]
C. 0
D. -1

Answer 1

Hint: In the above type of question we will have to use the trigonometric formulae and rearrange the given expression using trigonometric identity or formulae to get the result. Here we will square the both sides of the given condition and solve it further and with the use of the formula of \[\sin 2\theta =2\sin \theta \cos \theta \] we will get the result. We will also use the trigonometric identity which is \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\]. So, we will use the above formula and the identity to get the answer.

Complete step-by-step answer:

Here in the above question it is given that $\sin \theta +\cos \theta =1$. So, we will square it both side of equal and we get,
$\begin{align}
  & {{\left( \sin \theta +\cos \theta \right)}^{2}}={{1}^{2}} \\
 & \Rightarrow {{\sin }^{2}}\theta +{{\cos }^{2}}\theta +2\sin \theta \cos \theta =1 \\
\end{align}$

Now, in the above equation we will use the trigonometric identity \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\] and we get,
\[\begin{align}
  & \Rightarrow 1+2\sin \theta \cos \theta =1 \\
 & \Rightarrow 2\sin \theta \cos \theta =1-1=0 \\
\end{align}\]

Hence, we can see that the value of \[\sin 2\theta =2\sin \theta \cos \theta \]= 0.

Therefore, the correct option of the above question will be option C.

Note: Be careful while doing calculation because there is a chance that you might make mistakes during calculation and you will get the incorrect answer.
Also, remember the trigonometric formulae which are very helpful in these types of questions.
Also, remember the trigonometric identity , sometimes it makes your question easy.