Question

Find \cos\left(\dfrac{19\pi}{12}\right)cos( 12 19π ​ )cosine, left parenthesis, start fraction, 19, pi, divided by, 12, end fraction, right parenthesis exactly using an angle addition or subtraction formula.

Answer 1

The value of given expression is
`-(√(2)-√(6))/(4)`.

Explanation:

The given expression is

`\cos\left((19\pi)/(12)\right)`

The trigonometric ratios are not defined for
`(19\pi)/(12)`.

`(19\pi)/(12)` can be split into
`(5\pi)/(4)+(\pi)/(3)`.

`\cos\left((19\pi)/(12)\right)=\cos ((5\pi)/(4)+(\pi)/(3))`

Using the addition formula

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

`\cos ((5\pi)/(4)+(\pi)/(3))=\cos( (\pi)/(3))\cdot \cos ((5\pi)/(4))-\sin( (\pi)/(3))\cdot \sin ((5\pi)/(4))`

We know that,
`\cos((\pi)/(3))=(1)/(2)` and
`\sin ((\pi)/(3))=(√(3))/(2)`

`\cos\left((19\pi)/(12)\right)=(1)/(2)\cdot \cos ((5\pi)/(4))-(√(3))/(2)\cdot \sin ((5\pi)/(4))`

`(5\pi)/(4)` lies in third quadrant, by using reference angle properties,

`\cos((5\pi)/(4))=-\cos((\pi)/(4))=-(√(2))/(2)`

`\sin((5\pi)/(4))=-\sin((\pi)/(4))=-(√(2))/(2)`

`\cos\left((19\pi)/(12)\right)=(1)/(2)\cdot (-(√(2))/(2))-(√(3))/(2)\cdot (-(√(2))/(2))`

`\cos\left((19\pi)/(12)\right)=-(√(2))/(4)+(√(6))/(4)`

`\cos\left((19\pi)/(12)\right)=-((√(2)-√(6)))/(4)`

Therefore the value of given expression is
`-(√(2)-√(6))/(4)`.