how do you type sec^2(0) on a calculator?

Question

I press cos^-1 then ^2 then brack (o) but then it comes up with syntax error

Answer 1

Here is again one of the times to remind everyone of the unfortunate way in which mathematics, and in particular trigonometric functions, are written in today's systems.

$$\cos^{n}(x) = \begin{cases} [\cos(x)]^n & n\neq -1 \\ \arccos(x) & n=-1\end{cases}$$

That is to say, on calculators, and many papers, if you see $\cos^{-1}(x)$ they do not mean the multiplicative inverse, $\sec(x)=\frac{1}{\cos(x)}$, but instead they mean to say the inverse function such that $\cos^{-1}(\cos(x)) = x$.

To avoid ambiguity, it is highly recommended to never write $\cos^{-1}$ and instead either write $\sec, \frac{1}{\cos}$ or $\arccos$ depending on what you intend to use.

(the other trigonometric functions suffer the same unfortunate ambiguity)

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To calculate $\sec^2{x}$ on a graphing calculator, you should have the ability to write strings of functions with parenthesis and the like and can write it as:

$$1 / (\cos(x))^2$$

If you are using a scientific calculator, then you could push the buttons in the following order: inputnumber for $x$,$~~~~\cos,~~~~x^2,~~~~x^{-1}$

(depending on the model of the calculator either inputnumber$~~~~\cos$ will give the value for cos(input) or you will need to reverse the order of the button presses to $\cos~~~~~$inputnumber.)

Answer 2

$\cos^{-1}(x)$ does not mean $(\cos(x))^{-1}=\sec(x)$.

(Note that $\cos^2(x)$ does mean $(\cos(x))^2$. The notation is inconsistent, and there's no real reason for it except for the fact that whoever made up the notation did it in a stupid way.)

Instead, $\cos^{-1}(x)$ means $\arccos(x)$, the inverse of $\cos(x)$ (that is, if $\cos(x)=y$, then $\arccos(y)=x$, usually).

If you want to get $\sec^2(0)$, you should first notice that $\sec^2(0)=\dfrac1{\cos^2(0)}$. I would, then, first get $\cos(0)$, and then press the x^2 and 1/x buttons. (Or, on a graphing calculator, I would just type in 1/cos(0)^2 or something similar.

Answer 3

Hit these keys:

cos(0)*cos(0)
=
1/x

on a TI-type calc and you will get 1.0.

Answer 4

The correct key sequence depends on what calculator you are using. Some calculators require a very different key sequence than others for the same calculation. But a few observations are likely to apply to any calculator.

As an another answer has already pointed out, a key labeled "$\cos^{-1}$" does not perform a secant function. If there is no key labeled "sec" then you need some combination involving the "cos" key and the "$1/x$" key if there is one, or you have to divide $1$ by $\cos \theta$ instead of multiplying by $\sec \theta$, or if you just want $\sec \theta$ alone you can divide $1$ by $\cos \theta$.

For the "$^2$" part of $\sec^2 \theta$, most calculators that have an $x^2$ key require you to compute $\cos\theta$ first, and then use the $x^2$ key.

Answer 5

OK. If you have the old calculators, you don't have a fraction (hate those). For those, enter 1/(cos(0))^2.

For the new calculators (love those), you do ALPHAY=ENTERALPHA1>cos(0))^2