Question

the unit circle is shown. use the unit circle to find each value. cos(11π/6)= sin(11π/6)= tan(11π/6)=

Explanation:

Step1: Recall unit - circle trigonometry

On the unit - circle, for an angle $\theta$, $\cos\theta$ is the $x$ - coordinate of the point on the unit - circle corresponding to the angle $\theta$, $\sin\theta$ is the $y$ - coordinate of the point, and $\tan\theta=\frac{\sin\theta}{\cos\theta}$.
The angle $\theta = \frac{11\pi}{6}$ is equivalent to $330^{\circ}$.

Step2: Find $\cos\frac{11\pi}{6}$

Looking at the unit - circle, for $\theta=\frac{11\pi}{6}$, the $x$ - coordinate of the corresponding point is $\frac{\sqrt{3}}{2}$. So, $\cos\frac{11\pi}{6}=\frac{\sqrt{3}}{2}$.

Step3: Find $\sin\frac{11\pi}{6}$

For $\theta = \frac{11\pi}{6}$, the $y$ - coordinate of the corresponding point is $-\frac{1}{2}$. So, $\sin\frac{11\pi}{6}=-\frac{1}{2}$.

Step4: Find $\tan\frac{11\pi}{6}$

Since $\tan\theta=\frac{\sin\theta}{\cos\theta}$, substituting $\sin\frac{11\pi}{6}=-\frac{1}{2}$ and $\cos\frac{11\pi}{6}=\frac{\sqrt{3}}{2}$, we get $\tan\frac{11\pi}{6}=\frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}}=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}$.

Answer:

$\cos\frac{11\pi}{6}=\frac{\sqrt{3}}{2}$, $\sin\frac{11\pi}{6}=-\frac{1}{2}$, $\tan\frac{11\pi}{6}=-\frac{\sqrt{3}}{3}$