Question

How do you evaluate \[\sin \left( \dfrac{\pi }{3}-\dfrac{\pi }{4} \right)\]?

Answer 1

Hint: In this problem, we have to evaluate the given trigonometric expression. we should know that, to solve these types of problems, we have to know some trigonometric identities, rules, formulas and some degree values for sine and cosine. In this problem, we are going to use a trigonometric identity and find the degree values, to evaluate this problem.

Complete step by step solution:
We know that the given trigonometric expression is,
 \[\sin \left( \dfrac{\pi }{3}-\dfrac{\pi }{4} \right)\].
We know that the trigonometric identity to be used in this problem is,
\[\sin \left( A-B \right)=\sin A\cos B-\cos A\sin B\]
We can apply this identity in the given trigonometric expression, we get
\[\Rightarrow \sin \left( \dfrac{\pi }{3}-\dfrac{\pi }{4} \right)=\sin \left( \dfrac{\pi }{3} \right)\cos \left( \dfrac{\pi }{4} \right)-\cos \left( \dfrac{\pi }{3} \right)\sin \left( \dfrac{\pi }{4} \right)\] …….. (1)
 Now we can apply the degree values in the above step, we know that
\[\begin{align}
  & \sin \left( \dfrac{\pi }{3} \right)=\dfrac{\sqrt{3}}{2} \\
 & \cos \left( \dfrac{\pi }{4} \right)=\dfrac{1}{\sqrt{2}} \\
 & \cos \left( \dfrac{\pi }{3} \right)=\dfrac{1}{2} \\
 & \sin \left( \dfrac{\pi }{4} \right)=\dfrac{1}{\sqrt{2}} \\
\end{align}\]
Now we can substitute the above values in the step (1), we get
\[\begin{align}
  & \Rightarrow \sin \left( \dfrac{\pi }{3}-\dfrac{\pi }{4} \right)=\sin \left( \dfrac{\pi }{3} \right)\cos \left( \dfrac{\pi }{4} \right)-\cos \left( \dfrac{\pi }{3} \right)\sin \left( \dfrac{\pi }{4} \right) \\
 & \Rightarrow \sin \left( \dfrac{\pi }{3}-\dfrac{\pi }{4} \right)=\dfrac{\sqrt{3}}{2}\times \dfrac{1}{\sqrt{2}}-\dfrac{1}{2}\times \dfrac{1}{\sqrt{2}} \\
\end{align}\]
Now we can simplify the above step, to get the exact value
We know that we have same denominator, so we can subtract the two numerators with a single denominator, as it is identical, to get the value, we get
\[\Rightarrow \sin \left( \dfrac{\pi }{3}-\dfrac{\pi }{4} \right)=\dfrac{\sqrt{3}-1}{2\sqrt{2}}\]
Therefore, the value of \[\sin \left( \dfrac{\pi }{3}-\dfrac{\pi }{4} \right)=\dfrac{\sqrt{3}-1}{2\sqrt{2}}\].

Note: Students make mistakes while writing the correct formula or identity which is based on this given sum, which should be concentrated. We should know that, to solve these types of problems, we have to know some trigonometric identities, rules, formulas and some degree values for sine and cosine. We should also write the correct degree values, to get the final answer correct.