Evaluate tan(4.2 pi), giving an exact value.

Step-by-step solution

1. 1

   Use the period of tangent, which is $\pi$ and not $2\pi$. Unlike sine and cosine, tangent repeats every half turn:

   $\tan(\theta+k\pi)=\tan\theta\quad\text{for any integer }k$

   Using $2\pi$ here would still work but would leave more to reduce.

2. 2

   Split off whole multiples of $\pi$. Write $4.2=4+0.2$:

   $\tan(4.2\pi)=\tan(4\pi+0.2\pi)=\tan(0.2\pi)$

   since $4\pi$ is four whole periods.

3. 3

   Convert the remaining angle to a familiar form.

   $0.2\pi=\frac{\pi}{5}=36^\circ$

   so the problem reduces to finding $\tan 36^\circ$.

4. 4

   Quote the exact value for the pentagon angle. The angle $36^\circ$ is tied to the regular pentagon and the golden ratio, and its tangent has a closed radical form:

   $\tan\frac{\pi}{5}=\sqrt{5-2\sqrt5}$

5. 5

   Verify numerically. $\sqrt5\approx 2.236068$, so $5-2\sqrt5\approx 0.527864$ and its square root is $\approx 0.726543$. A direct evaluation gives $\tan(4.2\pi)\approx 0.7265425$ $\checkmark$. The value is positive, consistent with $\tfrac{\pi}{5}$ lying in the first quadrant.