Question

cos^-1(cos4pi/5) can someone help me with this one? And explain how they got the answer as well
Will fan and medal

Answer 1

Is cos^-1 arccosine (the inverse of the function) or 1/cos?

Answer 2

the inverse

Answer 3

That makes things nice ^_^ So the inverse of cosine does the opposite operation that cosine performs on a number. So whereas
\[cos \left( \frac{4 \pi}{5} \right) \approx -.809\]
if you perform an inverse function on a function, you just get the original input, so
\[ cos^{-1}\left( cos\left( \frac{4 \pi}{5} \right) \right) = \frac{4 \pi}{5}\]
They kinda cancel each other out.

Answer 4

If there's a negative in the parentheses does that change things? @AllTehMaffs

Answer 5

? like a negative before the cosine?
\[cos^{-1}\left( -cos \left( \frac{4 \pi}{5} \right) \right) \]
?

Answer 6

or before the -4pi/5 ?

Answer 7

Like for number 42. on my homework
cos^-1 [cos(-5pi/3)}

Answer 8

The output of the cos^-1 is always what's in the argument of the cosine that it's inverting, so the answer to that one would be (-5 pi / 3).

If it's before the cosine, then the answer is different because you can only invert a positive cosine. If you're not being quizzed on it though, don't worry about that part.

Answer 9

By definition, we have:
\[{\cos ^{ - 1}}\left( {\cos \left( {\frac{{4\pi }}{5}} \right)} \right) = \frac{{4\pi }}{5} + 2k\pi ,\quad k \in \mathbb{Z}\]

Answer 10

just to state the obvious by now \(\large \bf {
cos^{-1}[cos({\color{red}{ \theta}})]={\color{red}{ \theta}}
\\ \quad \\
sin^{-1}[sin({\color{red}{ \theta}})]={\color{red}{ \theta}}
\\ \quad \\
tan^{-1}[tan({\color{red}{ \theta}})]={\color{red}{ \theta}}
}\)