Question

How do you solve \[\cos x = 3\]?

Answer 1

Hint: We have to find the value of \[x\] such that it satisfies the given condition. We solve this question using the concept of the value of the range of a trigonometric function. We should have the knowledge of the possible range in which the value of a trigonometric function can exist. First, we will write the expression for the range of the cosine function and then check whether the value of the expression lies in the range of the cosine function or not. And then we will simplify the given expression such that we get the value of \[x\].

Complete step-by-step answer:
Given:
\[\cos x = 3\]
As, we know that the range for the cosine function is given as:
\[ - 1 \leqslant \cos x \leqslant 1\]
i.e. range is \[\left[ { - 1,1} \right]\].
As, we conclude that the maximum value which is possible for the cosine function, \[\cos x\] is \[1\]. And the given value for the cosine function, \[\cos x\] is \[3\].
Hence, the given function does not exist. As for no value of \[x\] the cosine function can have value other than the given range \[\left[ { - 1,1} \right]\] for the given function \[\cos x\].
Thus, the value of \[x\] does not exist for the given expression \[\cos x = 3\].

Note: The value of the range of a trigonometric function changes with the angle of the given function. We can check the value of the range of a function by putting different values of angles in the given function. The range defines for no angle the value of that function can produce the value outside the range.