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So I've been playing around with my calculator.

In degrees mode with my casio calculator, I typed, "$\arctan(1)$" and got an answer of 45.

In radian mode with the same parameter, I got, "0.785398..."

So I'm thinking like, "If you can convert degrees to radian in sin(x) by multiplying x with $\frac{180}{\pi}$, it should work with arctan", but when I tried in degrees mode with the parameter $\arctan(\frac{180}{\pi})$, it gave me "89.0001..." which is far from the correct value of "0.785398...".

Where did I went wrong?

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2 Answers 2

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In degrees mode, $\arctan(1) = 45^{\circ}$. So, if you want this in radians, you need to convert $45^{\circ}$ to radians, if you do so, you get $0.785$, as expected.

One way to think of this is that the thing which is in degrees/radians is the angle; i.e the thing which you input into the trigonometric functions $\sin(\cdot), \cos(\cdot), \tan(\cdot), \csc(\cdot), \sec(\cdot), \cot(\cdot)$, NOT their inverses. If you look at the inverse functions then it is the output which is an angle.

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The output of $\arctan(1)$ is the angle, not the input.

Put the calculator in radian mode and write $\frac{180}{\pi}\arctan(1)$ to see what you get.

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