20. use the figure to find the value of tan(27π/4). tan(27π/4) = enter your next step here

Explanation:

Step1: Simplify the angle

First, reduce $\frac{27\pi}{4}$. We know that $2\pi$ is one - full rotation. Divide $27$ by $8$ (since $2\pi=\frac{8\pi}{4}$). $27 = 4\times6+3$. So, $\frac{27\pi}{4}=6\pi+\frac{3\pi}{4}$. Since adding a multiple of $2\pi$ to an angle does not change the value of the trigonometric function, $\tan(\frac{27\pi}{4})=\tan(6\pi + \frac{3\pi}{4})=\tan(\frac{3\pi}{4})$.

Step2: Recall the tangent formula

The formula for $\tan\theta=\frac{\sin\theta}{\cos\theta}$. For $\theta = \frac{3\pi}{4}$, from the unit - circle, $\sin(\frac{3\pi}{4})=\frac{\sqrt{2}}{2}$ and $\cos(\frac{3\pi}{4})=-\frac{\sqrt{2}}{2}$.

Step3: Calculate the tangent value

$\tan(\frac{3\pi}{4})=\frac{\sin(\frac{3\pi}{4})}{\cos(\frac{3\pi}{4})}=\frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}=- 1$.

Answer:

$-1$