Answer:
[tex]tan(\frac{-2\pi}{3} ) =\sqrt{3} \approx1.732[/tex]
Step-by-step explanation:
First of all let's define the tangent function:
[tex]tan(\theta )=\frac{Opposite}{Adjacent} =\frac{sin(\theta)}{cos(\theta)}[/tex]
Now, let's define the standard angles, standard angles are those that have values that appear very often in everyday life. These angles are 30°=π/6, 45°=π/4, and 60°=π/3, and the angles 0°, 90°=π/2, 120°= 2π/3, 180°=π, 270°=3π/2, and 360°=2π. The latter, although not defined as 'standard', are also very common. Here are the values:
[tex]cos(0)=1\hspace{25}sin(0)=0\\cos(\frac{\pi}{6} )=\frac{\sqrt{3}}{2} \hspace{15}sin(\frac{\pi}{6})=\frac{1}{2} \\cos(\frac{\pi}{4} )=\frac{\sqrt{2}}{2} \hspace{15}sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}[/tex]
[tex]cos(\frac{\pi}{3})=\frac{1}{2}\hspace{28} sin(\frac{\pi}{3} )=\frac{\sqrt{3}}{2} \\cos(\frac{\pi}{2})=0\hspace{28} sin(\frac{\pi}{2} )=1}\\cos(\frac{2\pi}{3})=-\frac{1}{2}\hspace{15} sin(\frac{2\pi}{3} )=\frac{\sqrt{3}}{2}[/tex]
[tex]cos(\pi)=-1\hspace{25}sin(\pi)=0\\cos(\frac{3\pi}{2} )=0 \hspace{28}sin(\frac{3\pi}{2})=-1 \\cos(2\pi )=1 \hspace{27}sin(2\pi)=0[/tex]
Also you need to keep in mind that cosine function is an even function, and sine function is an odd function, that is:
[tex]cos(-\theta)=cos(\theta)\\\\sin(-\theta)=-sin(\theta)[/tex]
Using these definitions you are able to solve the problem:
[tex]tan(\frac{-2\pi}{3} ) =\frac{sin(\frac{-2\pi}{3} )}{cos(\frac{-2\pi}{3} )} = \frac{\frac{-\sqrt{3} }{2} }{\frac{-1}{2} } = \sqrt{3} \approx1.732[/tex]
