Given:
The angle is 240 degrees.
Required:
We need to find the six trigonometric functions.
Explanation:
Consider the sine function.
[tex]sin(240\degree)=sin(180+60)\degree[/tex][tex]=-sin(60)\degree[/tex][tex]=\frac{-\sqrt{3}}{2}[/tex][tex]sin(240\degree)=\frac{-\sqrt{3}}{2}[/tex]Consider the cosine function.
[tex]cos(240\degree)=cos(180+60)\degree[/tex][tex]=-cos(60)\degree[/tex][tex]=-\frac{1}{2}[/tex][tex]cos(240\degree)=-\frac{1}{2}[/tex]Consider the tan function.
[tex]tan(240)\degree=\frac{sin(240)\degree}{cos(240)\degree}[/tex][tex]Substitute\text{ }sin(240\degree)=\frac{-\sqrt{3}}{2}\text{ and }cos(240\degree)=-\frac{1}{2\text{ }}\text{ in the equation.}[/tex][tex]tan(240)\degree=\frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}}[/tex][tex]tan(240)\degree=-\frac{\sqrt{3}}{2}\times(-\frac{2}{1})[/tex][tex]tan(240)\degree=\sqrt{3}[/tex]Consider the cot function.
[tex]cot(240)\degree=\frac{cos(240)\degree}{sin(240)\degree}[/tex][tex]Substitute\text{ }sin(240\degree)=\frac{-\sqrt{3}}{2}\text{ and }cos(240\degree)=-\frac{1}{2\text{ }}\text{ in the equation.}[/tex][tex]cot(240)\degree=\frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}}[/tex][tex]cot(240)\degree=-\frac{1}{2}\times(-\frac{2}{\sqrt{3}})[/tex][tex]cot(240)\degree=\frac{1}{\sqrt{3}}[/tex]Consider the sec function.
[tex]sec(240)\degree=\frac{1}{cos(240)\degree}[/tex][tex]Substitute\text{ }cos(240\degree)=-\frac{1}{2\text{ }}\text{ in the equation.}[/tex][tex]sec(240)\degree=\frac{1}{-\frac{1}{2}}[/tex][tex]sec(240)\degree=1\times(-\frac{2}{1})[/tex][tex]sec(240)\degree=-2[/tex]Consider the csc function.
[tex]csc(240)\degree=\frac{1}{sin(240)\degree}[/tex][tex]Substitute\text{ }sin(240\degree)=\frac{-\sqrt{3}}{2}\text{ in the equation.}[/tex][tex]csc(240)\degree=\frac{1}{-\frac{\sqrt{3}}{2}}[/tex][tex]csc(240)\degree=1\times(\frac{-2}{\sqrt{3}})[/tex][tex]csc(240)\degree=\frac{-2}{\sqrt{3}}[/tex]Consider the angle in a graph:
Final answer:
[tex]sin(240\degree)=\frac{-\sqrt{3}}{2}[/tex][tex]cos(240\degree)=-\frac{1}{2}[/tex][tex]tan(240)\degree=\sqrt{3}[/tex][tex]cot(240)\degree=\frac{1}{\sqrt{3}}[/tex][tex]sec(240)\degree=-2[/tex][tex]csc(240)\degree=\frac{-2}{\sqrt{3}}[/tex]The angle lies in the third quadrant.
