Which of the following values can the function $y = \tan(x)$ take on if $\frac{\pi}{2} \lt x \lt \frac{3\pi}{4}$ holds

trigonometry

I have this mathematical problem, but can't seem to figure out reasoning behind this:

Which of the following values can the function $y = \tan(x)$ take on if $\frac{\pi}{2} \lt x \lt \frac{3\pi}{4}$ holds?

List of options to choose:

  • $-8$
  • $-\frac{1}{8}$
  • $0$
  • $\frac{1}{8}$
  • $8$

I knew that on this interval of unit circle, the value of $\tan(x)$ will be negative, since value of $\cos(x)$ is negative and $\sin(x)$ is positive here. So either $-8$ or $-\frac{1}{8}$ has to be the right answer. I choose $-\frac{1}{8}$, but the correct answer was $-8$. Why?

Best Answer

It's because you are in only half of the second quadrant, the half that is closer to the $y$ axis. This means in that 45 degree arc, the $x$ coordinate will be smaller than the $y$ coordinate in absolute value, and since $\tan(\theta)=\frac y x$ where $(x,y)$ is the point on the unit circle associated with $\theta$, in this area your numerator will be greater than your denominator, which means the magnitude will be greater than 1.

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