Question

Find an exact value. sine of negative eleven pi divided by twelve.

Answer 1

sin(-11pi/12) = -sin(11pi/12) = -sin(pi - pi/12) = -sin(pi/12) = -sin( (pi/6) / 2)

= - sqrt( (1-cos(pi/6) ) / 2) = -sqrt( (1-√3/2) / 2 ) = -(√3-1) / 2√2

Answer 2

The exact value is:

`\sin ((-11\pi)/(12))=-((√(3)-1)/(2√(2)))`

Explanation:

We are asked to find the value of:

`\sin ((-11\pi)/(12))`

We know that for any angle theta (θ) :

`\sin (-\theta)=-\sin \theta`

Hence, we get:

`\sin ((-11\pi)/(12))=-\sin ((11\pi)/(12))`

Hence,

`\sin ((-11\pi)/(12))=-\sin (\pi-(\pi)/(12))\\\\\\i.e.\\\\\\\sin ((-11\pi)/(12))=-\sin ((\pi)/(12))`

Since,

`\sin (\pi-\theta)=\sin \theta`

Also,

`\sin ((\pi)/(12))=\sin ((\pi)/(3)-(\pi)/(4))`

Hence,

on using the formula:

`\sin (A-B)=\sin A\cdot \cos B-\cos A\cdot \sin B`

Hence, we get:

`\sin ((\pi)/(12))=\sin ((\pi)/(3))\cos ((\pi)/(4))-\cos ((\pi)/(3))\sin ((\pi)/(4))\\\\\\\sin ((\pi)/(12))=(√(3))/(2)\cdot (1)/(√(2))-(1)/(2)\cdot (1)/(√(2))\\\\\\\sin ((\pi)/(12))=(√(3)-1)/(2√(2))`

Hence, the value is:

`\sin ((-11\pi)/(12))=-((√(3)-1)/(2√(2)))`