Determine whether each of the following is possible for a real number m: cos(m) = 2, tan(m) = 2, cot(0) = m, csc(pi) = m, sin(m) = -1.1, arcsec(m) = pi/2, cot(pi) = m, sin(m) = -0.9.

Step-by-step solution

1. 1

   Sort the eight statements into two kinds. In (a), (b), (e) and (h) the unknown is inside the function, so the question is whether the given output lies in the function's range. In (c), (d) and (g) the unknown is the output, so the question is whether the function is even defined at that input. Part (f) is about the range of an inverse function.

2. 2

   Use the range of sine and cosine for (a), (e) and (h). For every real $m$,

   $-1\le\sin m\le 1,\qquad -1\le\cos m\le 1.$

   So $\cos(m)=2$ is not possible and $\sin(m)=-1.1$ is not possible, while $\sin(m)=-0.9$ is possible (for instance $m=\arcsin(-0.9)\approx-1.1198$).

3. 3

   Use the range of tangent for (b). $\tan$ takes every real value on each branch, so $\tan(m)=2$ is possible; $m=\arctan 2\approx 1.1071$ works, as do $m\approx 1.1071+k\pi$.

4. 4

   Check where cotangent and cosecant blow up, for (c), (d) and (g). Both have $\sin$ in the denominator:

   $\cot\theta=\frac{\cos\theta}{\sin\theta},\qquad \csc\theta=\frac{1}{\sin\theta}.$

   Since $\sin 0=0$ and $\sin\pi=0$, all three of $\cot(0)$, $\csc(\pi)$ and $\cot(\pi)$ are undefined — they are $\tfrac10$, $\tfrac10$ and $\tfrac{-1}{0}$. No real $m$ can equal an undefined expression, so (c), (d) and (g) are not possible.

5. 5

   Check the range of inverse secant for (f). The inverse secant is defined with range $\left[0,\tfrac{\pi}{2}\right)\cup\left(\tfrac{\pi}{2},\pi\right]$ — the value $\tfrac{\pi}{2}$ is deliberately excluded, because $\sec\left(\tfrac{\pi}{2}\right)=\tfrac{1}{\cos(\pi/2)}=\tfrac10$ is undefined. So $\sec^{-1}(m)=\tfrac{\pi}{2}$ is not possible.

6. 6

   Collect the verdicts. Possible: (b) and (h). Not possible: (a), (c), (d), (e), (f), (g). The pattern worth remembering is that bounded functions fail on size, and reciprocal functions fail wherever their denominator hits zero.