Question

Find the value of $\sin \left( 270{}^\circ \right)$.

Answer 1

Hint: The value of $\sin \theta$ is negative in the third quadrant. Also, we can write $\sin \left( 270{}^\circ \right)$ in the form of $\sin \left( 180{}^\circ +\theta \right)$. We will then apply the formula of $\sin \left( \pi +\theta \right)=-\sin \theta$ to find the value of $\sin \left( 270{}^\circ \right)$.

Complete step-by-step answer:
It is given in the question that we have to find the value of $\sin \left( 270{}^\circ \right)$. We know that the value of $\sin \theta$ increases with the increase in the value of $\theta$. We also know that the value of $\sin \theta$ is negative in the third quadrant. We also know that, $\sin \left( \pi +\theta \right)=-\sin \theta$. So, we will use this formula to find the value of $\sin \left( 270{}^\circ \right)$.

Now, we know that we can also write $\sin \left( 270{}^\circ \right)$ as $\sin \left( 180{}^\circ +90{}^\circ \right)$. So, applying the formula, we will get,
$\sin \left( 180{}^\circ +90{}^\circ \right)=-\sin 90{}^\circ$
We know that the value of $\sin 90{}^\circ =1$, so $-\sin 90{}^\circ =-1$. So, we by applying that, we will get,
$\sin \left( 180{}^\circ +90{}^\circ \right)=-1$ or $\sin \left( 270{}^\circ \right)=-1$.
Therefore, we get the value of $\sin \left( 270{}^\circ \right)=-1$.

Note: The students must remember the trigonometric formulas to answer such questions. They should not get confused with the formulas and miss out on any signs, like they should not write the formula as, $\sin \left( \pi +\theta \right)=\sin \theta$ as that would lead to an incorrect answer. We can also solve this using an alternate method. We can use the formula, $\sin \left( 2\pi -\theta \right)=-\sin \theta$. So, we can express $\sin \left( 270{}^\circ \right)$ as $\sin \left( 360{}^\circ -90{}^\circ \right)=-\sin 90{}^\circ =-1$.