Question

Find an exact value cos(19pi/12)

Answer 1

19 π / 12 = 19 * 180° / 12 = 285 °
cos ( 19 π / 12 ) = cos 285° = cos ( - 75° ) = cos 75°
cos 75° = cos ( 45° + 30° ) = cos 45° cos 30° - sin 45° sin 30° =
= ( √2 / 2 · √3/2 ) - ( √2 / 2 · 1/2 ) = √6/4 - √2/4 =
= ( √6 - √2 )/ 4

Answer 2

`cos\left ( (9\pi)/(12)\right )=\frac{\sqrt{2-√(3)}}{2}`

Explanation:

We know that

cos 2x = 2cos²x - 1

So we have

`cosx=\sqrt{(1+cos2x)/(2)}`

Substituting
`x=(19\pi)/(12)`

`cos\left ( (19\pi)/(12)\right )=\sqrt{(1+cos2\left ( (19\pi)/(12)\right ))/(2)}\\\\cos\left ( (19\pi)/(12)\right )=\sqrt{(1+cos\left ( (19\pi)/(6)\right ))/(2)}\\\\cos\left ( (9\pi)/(12)\right )=\sqrt{(1+cos570)/(2)}\\\\cos\left ( (9\pi)/(12)\right )=\sqrt{(1-(√(3))/(2))/(2)}\\\\cos\left ( (9\pi)/(12)\right )=\sqrt{(2-√(3))/(4)}\\\\cos\left ( (9\pi)/(12)\right )=\frac{\sqrt{2-√(3)}}{2}`

`cos\left ( (9\pi)/(12)\right )=\frac{\sqrt{2-√(3)}}{2}`