Question

Find \sin\left(\dfrac{7\pi}{12}\right)sin( 12 7π ​ )sine, (, start fraction, 7, pi, divided by, 12, end fraction, )exactly using an angle addition or subtraction formula.'

Answer 1

`(\sqrt 6 +\sqrt 2)/(4)`

Explanation:

Use the addition formula
`\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)` with x=4π/12 and y=3π/12:

`\sin((7\pi)/(12))&=\sin((4\pi)/(12)+(3\pi)/(12))=\sin((4\pi)/(12))\cos((3\pi)/(12))+\cos((4\pi)/(12))\sin((3\pi)/(12))=\sin((pi)/(3))\cos((\pi)/(4))+\cos((\pi)/(3))\sin((\pi)/(4))=(\sqrt 3)/(2)\cdot(\sqrt 2)/(2)+(1)/(2)\cdot(\sqrt 2)/(2)=(\sqrt 6 +\sqrt 2)/(4)`