Given: cosθ= -4/5 , sin x = -12/13 , θ is in the third quadrant, and x is in the fourth quadrant; evaluate tan 1/2 θ

answer choices:

A. -3

B. 3

C. 1/3

Step-by-step explanation:

If you're looking for what the half angle of the tangent of theta is, I'm a bit confused as to why you think the angle in the 4th quadrant, x, is relevant.  But maybe you don't know it isn't and it's a "trick" to throw you off.  Hmm...

Anyways, the half angle identity for tangent is

There are actually 3 identities for the tangent of a half angle, but this one works just as well as either of the others do, so I'm going with this one.  

If theta is in QIII, the value of -4 goes along the x axis and the hypotenuse is 5.  That makes the missing side, by Pythagorean's Theorem, -3.  Filling in our formula:

which simplifies a bit to

 and a bit more to

Bring up the lower fraction and flip it to divide to get

which of course simplifies to

-3.  Choice A.

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