Question

How do you solve \[\cos x = 1\]?

Answer 1

Hint: In the given question, we have been given a trigonometric function. First, we are going to transform the equation into \[x = {\cos ^{ - 1}}\left( 1 \right)\]. This trigonometric function is raised to some negative power. Also, this trigonometric function has a constant as its argument. We just need to know what this negative power means. Any trigonometric function raised to this negative power means that it is the inverse of that trigonometric function. So, we have to find the inverse of the given trigonometric function. By finding the inverse it means that we have to find the angle which when put into the given trigonometric function, yields the same value as is given in the inverse of the trigonometric function.

Complete step-by-step answer:
The given trigonometric function is \[x = {\cos ^{ - 1}}\left( 1 \right)\]. When a trigonometric function is raised to a negative power, it means that we have to find the inverse of the given trigonometric function. By finding the inverse it means that we have to calculate the angle which gives the value which is inside the given inverse of the trigonometric function, which is 0.
So, \[x = {\cos ^{ - 1}}\left( 1 \right) = 0^\circ \]
Here, we got the solution for just one value, we have to find for the whole range, so, we are going to apply the formula for general solution of the cosine, which is,
\[\cos \theta = \cos \alpha \]
\[ \Rightarrow \theta = 2n\pi + \alpha \]
So, for this question, we have,
\[{\cos ^{ - 1}}\left( 1 \right) = 2n\pi + 0 = 2n\pi \]
Hence, the value of \[x\] is \[2n\pi \].

Note: In the given question, we had to find the value of the angle of a given trigonometric expression. This expression had just one basic trigonometric function which was equal to a value. But some students make mistakes when writing the angle in the form of pi. They do not know the value of the constant in terms of angle or forget to divide the conversion factor while changing the angle in degrees to the angle in pi. So, it is important that we know the value of pi and do not forget to correctly change it into the other form.