Q: What is confusing $\tan$ asymptotes with zeros?

$\tan(x)$ is zero at $x = 0, \pi, 2\pi, \ldots$ (where $\sin = 0$) and has asymptotes at $x = \pi/2, 3\pi/2, \ldots$ (where $\cos = 0$).

Question 1

What is $y = \sin $?

Accepted Answer

Wave with amplitude 1, period $2\pi$, $y$-intercept 0. Maxima at $x = \pi/2 + 2\pi k$, minima at $x = 3\pi/2 + 2\pi k$, zeros at $x = \pi k$.

Question 2

What is $y = \cos $?

Accepted Answer

Same shape as $\sin$ but shifted: $\cos(x) = \sin(x + \pi/2)$. Amplitude 1, period $2\pi$, $y$-intercept 1.

Question 3

What is $y = 	an $?

Accepted Answer

Period $\pi$. Vertical asymptotes at $x = \pi/2 + \pi k$. Zero at $x = \pi k$.

Question 4

What is calculator in degrees instead of radians?

Accepted Answer

VCE Methods uses radians. Check mode.

Question 5

What is missing solutions?

Accepted Answer

$\sin(x) = 1/2$ has two solutions per period (in the first and second quadrants), not one.

Question 6

What is extending range incorrectly?

Accepted Answer

When solving $\sin(2x) = c$ for $x \in [0, 2\pi]$, the composite $2x$ ranges over $[0, 4\pi]$, so look for solutions in that larger interval before dividing.

Question 7

What is treating $\sin^{-1} $ as $1/\sin $?

Accepted Answer

$\sin^{-1}$ means the inverse function (arcsin), not the reciprocal $\csc$.

Question 8

What is confusing $	an$ asymptotes with zeros?

Accepted Answer

$	an(x)$ is zero at $x = 0, \pi, 2\pi, \ldots$ (where $\sin = 0$) and has asymptotes at $x = \pi/2, 3\pi/2, \ldots$ (where $\cos = 0$).