Question

What does \[\tan 2\pi \] equal?

Answer 1

Hint: We are given a question based on trigonometric function and we are asked to find the value of the given trigonometric function for a specific angle. We have the tangent function and so we can calculate the value of tangent by writing the given function in terms of sine and cosine function, which is, \[\tan \theta =\dfrac{\sin \theta }{\cos \theta }\] and then substituting values in that expression. Solving the expression further, we will get the value of the given function.

Complete step by step solution:
According to the given question, we are given a trigonometric function – tangent function and we are asked to find the value for the given value of the angle.
The function we have is,
\[\tan 2\pi \]----(1)
We will first write the tangent function in terms of sine and cosine function and then we will substitute the values and get the value of the given function.
We know that, tangent function is the ratio of sine and cosine function which is represented as, \[\tan \theta =\dfrac{\sin \theta }{\cos \theta }\], so we get the new expression as,
\[\tan 2\pi =\dfrac{\sin 2\pi }{\cos 2\pi }\]
Now, we will put the values of the sine and cosine function at \[2\pi \]. \[2\pi \], if reduced using the period of trigonometry is similar to 0 degree, that is,
\[\sin 2\pi =0\] and \[\cos 2\pi =1\]
We get the new expression as,
\[\Rightarrow \tan 2\pi =\dfrac{0}{1}=0\]
Therefore, the value of \[\tan 2\pi =0\].

Note: We can also solve the above given function using the period of trigonometry, \[2\pi \] refers to one complete circle and it coincides with the 0 degree. So, we can write the given function as,
\[\tan 2\pi =\tan {{0}^{\circ }}\]
And we know that, \[\tan {{0}^{\circ }}=0\]
And so we have,
\[\Rightarrow \tan 2\pi =0\]
Therefore, the value of \[\tan 2\pi =0\].