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Question: Find the value of \(\sin \left( {\dfrac{{ - 11\pi }}{3}} \right)\)?...

Find the value of sin⁡(−11π3)\sin \left( {\dfrac{{ - 11\pi }}{3}} \right)?

Explanation

Solution

In this question we will try to find the value of sin⁡(−11π3)\sin \left( {\dfrac{{ - 11\pi }}{3}} \right). First, we will take this negative sign out of the bracket by using this identity sin⁡(−θ)=−sin⁡θ\sin ( - \theta ) = - \sin \theta . Then we will break 11π3\dfrac{{11\pi }}{3} in [4π−π3]\left[ {4\pi - \dfrac{\pi }{3}} \right] form. Again, we know that sin⁡(4π−θ)=−sin⁡θ\sin (4\pi - \theta ) = - \sin \theta , so we will use this identity to solve this question further.

Complete step-by-step answer:
Given,
sin⁡(−11π3)\sin \left( {\dfrac{{ - 11\pi }}{3}} \right)
=−sin⁡(11π3)= - \sin \left( {\dfrac{{11\pi }}{3}} \right) (As we know that sin⁡(−θ)=−sin⁡θ\sin ( - \theta ) = - \sin \theta )
=−sin⁡[4π−π3]= - \sin \left[ {4\pi - \dfrac{\pi }{3}} \right]
=−[−sin⁡π3]= - \left[ { - \sin \dfrac{\pi }{3}} \right] (As we know that sin⁡(4π−θ)=−sin⁡θ\sin (4\pi - \theta ) = - \sin \theta )
=sin⁡(π3)= \sin \left( {\dfrac{\pi }{3}} \right)
=32= \dfrac{{\sqrt 3 }}{2}

Additional Information: Radian measure is an alternative way to measure angles. Instead of dividing the circle into an arbitrary number of parts, we look at the length of the arc that subtends the angle. We measure the angle based on this length as a ratio to the radius. One radian is defined as the angle where the length of the arc equals the length of the radius. If we traverse the circle completely, we’ll have travelled the length of the circumference. The radian is a unit of measure for angles used mainly in trigonometry. It is used instead of degrees. Whereas a full circle is 360 degrees, a full circle is just over 6 radians.

Note: In trigonometry, we use pi (π\pi ) for 180 degrees to represent the angle in radians. Hence, sinπ\pi is equal to sin 180 or sin π\pi = 0. In trigonometry, the exact value of sin 180 is 0 as well as the value of sin 0 is equal to zero. The value of cos 180 degrees is -1. A 30-degree angle is equivalent to π6\dfrac{\pi }{6}radians. A 45-degree angle is equivalent to π4\dfrac{\pi }{4} radians. A 60-degree angle is equivalent toπ3\dfrac{\pi }{3} radians. A 90-degree angle is equivalent to π2\dfrac{\pi }{2} radians.