Question

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

Answer 1

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

`tan((\theta)/(2))=(sin\theta)/(1+cos\theta)`

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:

`tan((\theta)/(2))=(-(3)/(5) )/(1+(-(4)/(5)) )` which simplifies a bit to

`tan((\theta)/(2))=(-(3)/(5) )/((5)/(5) -(4)/(5) )` and a bit more to

`tan((\theta)/(2))=(-(3)/(5) )/((1)/(5) )`

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

`tan((\theta)/(2))=-(3)/(5)*(5)/(1)` which of course simplifies to

-3. Choice A.