Answer :
Answer:
[tex]\text{sin}(\frac{4\pi}{3})=-\frac{\sqrt{3}}{2}[/tex]
[tex]\text{csc}(\frac{4\pi}{3})=-\frac{2\sqrt{3}}{3}[/tex]
[tex]\text{cos}(\frac{4\pi}{3})=-\frac{1}{2}[/tex]
[tex]\text{sec}(\frac{4\pi}{3})=-2[/tex]
[tex]\text{tan}(\frac{4\pi}{3})=\sqrt{3}[/tex]
[tex]\text{cot}(\frac{4\pi}{3})=\frac{\sqrt{3}}{3}[/tex]
Step-by-step explanation:
We are asked to find the exact trigonometric ratio for given angle [tex]x=\frac{4\pi}{3}[/tex].
[tex]\text{sin}(x)=\text{sin}(\frac{4\pi}{3})[/tex]
[tex]\text{sin}(\frac{4\pi}{3})=\text{sin}(\pi+\frac{\pi}{3})[/tex]
Using summation identity, we will get:
[tex]\text{sin}(\pi+\frac{\pi}{3})=\text{sin}(\pi)\text{cos}(\frac{\pi}{3})+\text{cos}(\pi)\text{sin}(\frac{\pi}{3})[/tex]
[tex]\text{sin}(\pi+\frac{\pi}{3})=(0)\frac{1}{2}+(-1)(\frac{\sqrt{3}}{2})[/tex]
[tex]\text{sin}(\pi+\frac{\pi}{3})=0+-\frac{\sqrt{3}}{2}[/tex]
[tex]\text{sin}(\frac{4\pi}{3})=-\frac{\sqrt{3}}{2}[/tex]
Let us find [tex]\text{csc}(x)=\text{csc}(\frac{4\pi}{3})[/tex]
We will use identity [tex]\text{csc}(x)=\frac{1}{\text{sin}(x)}[/tex]
[tex]\text{csc}(\frac{4\pi}{3})=\frac{1}{\text{sin}(\frac{4\pi}{3})}=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}[/tex]
Now, we will solve for cos(x).
[tex]\text{cos}(\frac{4\pi}{3})=\text{cos}(\pi+\frac{\pi}{3})[/tex]
[tex]\text{cos}(\pi+\frac{\pi}{3})=\text{cos}(\pi)\text{cos}(\frac{\pi}{3})-\text{sin}(\pi)\text{sin}(\frac{\pi}{3})[/tex]
[tex]\text{cos}(\pi+\frac{\pi}{3})=(-1)\frac{1}{2}-(0)(\frac{\sqrt{3}}{2})[/tex]
[tex]\text{cos}(\pi+\frac{\pi}{3})=-\frac{1}{2}-0[/tex]
[tex]\text{cos}(\frac{4\pi}{3})=-\frac{1}{2}[/tex]
Let us find sec(x).
We will use identity [tex]\text{sec}(x)=\frac{1}{\text{cos}(x)}[/tex]
[tex]\text{sec}(\frac{4\pi}{3})=\frac{1}{\text{cos}(\frac{4\pi}{3})}=\frac{1}{-\frac{1}{2}}=-2[/tex]
Let us find tan(x).
We will use identity [tex]\text{tan}(x)=\frac{\text{sin}(x)}{\text{cos}(x)}[/tex].
[tex]\text{tan}(\frac{4\pi}{3})=\frac{\text{sin}(\frac{4\pi}{3})}{\text{cos}(\frac{4\pi}{3})}[/tex]
[tex]\text{tan}(\frac{4\pi}{3})=\frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}}}=\frac{\sqrt{3}*2}{2*1}=\sqrt{3}[/tex]
Let us find cot(x).
We will use identity [tex]\text{cot}(x)=\frac{1}{\text{tan}(x)}[/tex].
[tex]\text{cot}(\frac{4\pi}{3})=\frac{1}{\text{tan}(\frac{4\pi}{3})}[/tex]
[tex]\text{cot}(\frac{4\pi}{3})=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}[/tex]