Question

How do you know $\sin 30{}^\circ =\sin 150{}^\circ$?

Answer 1

In this problem we need to check whether the given condition is correct or not and if it is correct how can we prove that. In the given equation we can observe that the given equation has trigonometric ratios and we will consider those ratios individually. From a trigonometric table we can have the value of $\sin 30{}^\circ$. Now we will consider the value $\sin 150{}^\circ$, we will calculate the value of $\sin 150{}^\circ$ by using all silver tea cups methods. Now we will compare both the values to get the required result.

Complete step-by-step solution:
Given that $\sin 30{}^\circ =\sin 150{}^\circ$.
In the above equation we can observe the trigonometric ratio $\sin$ and the values $\sin 30{}^\circ$, $\sin 150{}^\circ$.
From the trigonometric table we have the value of $\sin 30{}^\circ$ as $\sin 30{}^\circ =\dfrac{1}{2}$.
Considering the value $\sin 150{}^\circ$.
We can write the angle $150{}^\circ$ as $180{}^\circ -30{}^\circ$. So, the value $\sin 150{}^\circ$ can be written as
$\sin 150{}^\circ =\sin \left( 180{}^\circ -30{}^\circ \right)$
We have the trigonometric formula $\sin \left( 180{}^\circ -\theta \right)=\sin \theta$, then the above equation is modified as
$\sin 150{}^\circ =\sin 30{}^\circ$
We have the value $\sin 30{}^\circ =\dfrac{1}{2}$, substituting this value in the above equation, then we will get
$\sin 150{}^\circ =\dfrac{1}{2}$
From the above two values we can write that $\sin 30{}^\circ =\sin 150{}^\circ =\dfrac{1}{2}$.

Note: For this problem there is no need to find the values in fact while calculating the value of $\sin 150{}^\circ$ we get the equation $\sin 150{}^\circ =\sin 30{}^\circ$ which is our required solution. We can also stop our solution when we get the equation $\sin 150{}^\circ =\sin 30{}^\circ$ while calculating the value of $\sin 150{}^\circ$ even without calculating the value of $\sin 30{}^\circ$.