When pi/2 < theta < 3pi/4, which of the following could possible be tan of theta? I know the answer is -8, I just don't understand how to get that. Could someone please help explain this to me?

Question

When pi/2 < theta < 3pi/4, which of the following could possible be tan of theta? I know the answer is -8, I just don't understand how to get that. Could someone please help explain this to me?

This was a multiple choice question and the answer choices were -8, -1/8, 0, 1/8, and 8.

Answer 1

Hi Ford,

Let's first think of what the consequence of π/2 < θ < 3π/4 is for the tan(θ) function:

This restriction on θ limits us to a region in the second quadrant (in degrees, this is between 90 degrees and 135 degrees)

The value of tan(θ) in this region is ALWAYS negative because tan(θ) = sin(θ) / cos(θ) and we can see that in the second quadrant, sin(θ) is always positive while cos(θ) is always negative, thus giving us a negative sign for tangent when we divide sin(θ) by cos (θ)

Ok, so we determined that the tangent of θ must be negative so that limits our choices to -8 or -1/8 at this point

Now, notice that in this region (π/2 < θ < 3π/4 or 90 degrees and 135 degrees), the magnitude of sin(θ) will always be greater than cos(θ) for any allowed angle. This means that tan(θ) (which is again = sin(θ) / cos(θ)) must be greater than 1 in magnitude (so NOT a fraction of one)

That is why it is -8

Answer 2