Question

What is the inverse cosine of \[\dfrac{1}{3}\] ?

Answer 1

Hint: Now to know the inverse cosine of any number that is find the angle at which we get the answer. For example we can say that the inverse cosine of 1 is \[0{}^\circ \] and \[360{}^\circ \] . Therefore we now need to find the angle at which we get the cosine to be \[\dfrac{1}{3}\] . Now we know that the cosine range is only from \[\left[ -1,1 \right]\] and the domain is infinite, therefore similarly for inverse of cosine we will need to make sure that is domain of inverse cosine is only from \[\left[ -1,1 \right]\] or else the angle wouldn’t exist.

Complete step-by-step answer:
Now here we will start by check if the number whose inverse cosine we need to find is allowed or not ( if its inverse cosine exists or not)
Now we can see that \[\dfrac{1}{3}\] that is \[0.3333\] lies in the range of \[\left[ -1,1 \right]\] therefore we can say that its cosine function does exist. Now to find its inverse cosine function we can write it as the expression that
\[\cos \theta =\dfrac{1}{3}\]
Now
\[\theta ={{\cos }^{-1}}\dfrac{1}{3}\]
Since we know that \[\dfrac{1}{3}\] is not the part of any special triangle therefore we can’t directly find its angle and we must use a trigonometric table or a calculator whatever is available to you to find the answer of the angle. On doing that we find that the angle will be
\[\theta \approx 70.53{}^\circ \]
Now we know that cosine is positive in both first quadrant and fourth quadrant therefore using this logic we can find the other angle where the inverse cosine exists that is
\[\theta \approx 360-70.53\]
Therefore
\[\theta \approx 289.47{}^\circ \]
Therefore the inverse cosine of \[\dfrac{1}{3}\] are found to be \[\theta \approx 70.53{}^\circ ,289.47{}^\circ \]

Note: There is always an infinite range for inverse cosine because of cosine being a periodic function. Therefore to limit it we will only take values from \[\left[ 0,2\pi \right]\] to limit the answers because after that the function will anyways be periodic and repeating.