Question

What is the exact value of tan
(19pie/12)

Answer 1

`\displaystyle -√(3) - 2`

Step-by-step explanation:

If you recall the unit circle [from the polar graph], you would have no trouble at all figuring this out, but sinse you have trouble, do not worry about it. So, here is what you should have realised:

`\displaystyle \boxed{1(1)/(2)\pi} = (18)/(12)\pi`

If you did not notise,
`\displaystyle 1(7)/(12)\pi,`or
`\displaystyle (19)/(12)\pi,`is just
`\displaystyle (\pi)/(12)`more than
`\displaystyle 1(1)/(2)\pi,`which means the exact value will either be a radical, followed by an integer or just a radical. In this case, accourding to the unit circle, you will have a radical, followed by an integer, and that will be this:

`\displaystyle (sin\:1(7)/(12)\pi)/(cos\: 1(7)/(12)\pi) = tan\:1(7)/(12)\pi \\ \\ \boxed{-√(3) - 2} = (sin\:1(7)/(12)\pi)/(cos\: 1(7)/(12)\pi)`

It is all about memorisation of the unit circle, which I know is difficult, but you will get used to it soon.

*Now, if you had to find
`\displaystyle cos\:(7)/(12)\pi,`then the exact value would be the OPPOCITE of the exact value of
`\displaystyle sin\:(7)/(12)\pi,`which is
`\displaystyle -(√(2) + √(6))/(4),`because you would be crossing into the second quadrant where the x-coordinates are negative, accourding to both the cartesian and polar graphs. In addition, sinse you ALREADY have both values for sine and cosine [accourding to your PREVIOUS QUESTION ASKED], all you need to do is divide both values in the order given above to arrive at the exact value for
`\displaystyle tan\:1(7)/(12)\pi.`

I am joyous to assist you at any time.