# How do you plug the inverse csc (-7/3) into the calculator?

Jul 28, 2015

Use the inverse sine function.

#### Explanation:

${\csc}^{-} 1 \left(- \frac{7}{3}\right)$ is a number between $- \frac{\pi}{2}$ and $\frac{\pi}{2}$ whose cosecant is $- \frac{7}{3}$).

That means the sine is $- \frac{3}{7}$.

${\csc}^{-} 1 \left(- \frac{7}{3}\right) = {\sin}^{-} 1 \left(- \frac{3}{7}\right)$.

In general,
because $\csc \theta = \frac{1}{\sin} \theta$,

we have: ${\csc}^{-} 1 x = {\sin}^{-} 1 \left(\frac{1}{x}\right)$

Note: we use a similar idea to find inverse secant and inverse cotangent on many calculators.

${\sec}^{-} 1 x = {\cos}^{-} 1 \left(\frac{1}{x}\right)$

and

${\cot}^{-} 1 x = {\tan}^{-} 1 \left(\frac{1}{x}\right)$

(The last works if we take the range of ${\cot}^{-} 1$ to be the same as that of ${\tan}^{-} 1$ -- a popular choice in this computer age.
The description is a bit more complicated if we take the range of ${\cot}^{-} 1 x$ to be the same as that of ${\cos}^{-} 1$ -- a traditional choice that makes the inverse cotangent continuous.)

Jul 28, 2015

Plug in calculator to find csc(-7/3)

Ans: arc x = -25.37 deg

#### Explanation:

Order of plug in:

(7) (:) (3) (=) (-) (1/x) (2nd) (sin)

The answer will be $x = - 25.37$ deg