Question

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

Answer 1

Answer:

$\displaystyle -\sqrt{3} - 2$

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:

$\displaystyle \boxed{1\frac{1}{2}\pi} = \frac{18}{12}\pi$

If you did not notise, $\displaystyle 1\frac{7}{12}\pi,$or $\displaystyle \frac{19}{12}\pi,$is just $\displaystyle \frac{\pi}{12}$more than $\displaystyle 1\frac{1}{2}\pi,$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:

$\displaystyle \frac{sin\:1\frac{7}{12}\pi}{cos\: 1\frac{7}{12}\pi} = tan\:1\frac{7}{12}\pi \\ \\ \boxed{-\sqrt{3} - 2} = \frac{sin\:1\frac{7}{12}\pi}{cos\: 1\frac{7}{12}\pi}$

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 $\displaystyle cos\:\frac{7}{12}\pi,$then the exact value would be the OPPOCITE of the exact value of $\displaystyle sin\:\frac{7}{12}\pi,$which is $\displaystyle -\frac{\sqrt{2} + \sqrt{6}}{4},$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 $\displaystyle tan\:1\frac{7}{12}\pi.$

I am joyous to assist you at any time.

Answer 2

1 + 2V3+3
-2
4 + 2V3
-2
dividing each term by -2 as a final step,
=-2 - V3