In my calculator (TI-84), there are only $sin, cos,$ and $tan$ commands (and inverse sin, inverse cos, inverse tan). I had a question that was as follows:
Calculate $csc(2.85)$ in which I was permitted use of the TI-84 calculator.
As $csc$ is the inverse of $sin$, I wrote $csc(2.85) = 1/sin(2.85)$, which came out to be $3.4785,$ which I believe to be the right answer. I wanted to take $csc^{-1}$ of $3.4785$ to check my answer, however when I did so, the answer came out to be $0.29159$ rather than $2.85.$
I checked with other random numbers that taking the inverse of the answer should give the correct number of radians back. Where did I go wrong?
$\endgroup$ 62 Answers
$\begingroup$$y=\csc(x)$ is a function, so there is only one solution to $\csc(2.85) = 3.4785$. However, if you take the inverse of the cosecant function, you will obtain the graph:
One can see that a vertical line at $x=3.4785$ will obtain a value just a bit bigger than $0$ and another value that is just a bit smaller than $\pi$. In fact, there will be an infinite number of values if the graph is $y=\csc^{-1}(x)$ because the graph is periodic in the vertical direction, that is, if the graph is not bounded by $y=\pm \frac\pi 2$. Hence, the general equation for the solution set is $$y=\csc^{-1}(x)+2n \pi \quad \text{or} \quad \pi - \csc^{-1}(x)+2n \pi \quad, \ n \in \mathbb{Z}$$ In this case specifically, $$y=\csc^{-1}(3.4785)+2n \pi \quad \text{or} \quad \pi - \csc^{-1}(3.4785)+2n \pi \quad, \ n \in \mathbb{Z}$$ $$\Rightarrow y=0.29159 \quad \text{or}\quad \pi - 0.29159 = 2.85$$
$\endgroup$ 0 $\begingroup$$\csc(x)=\frac{1}{\sin(x)}$
So $\csc(2.85)=\frac{1}{\sin(2.85)}$
Recall that $\sin(x)=\sin(\pi-x)$
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