Question

What are the 6 trig functions of a 240 degree angle. Sketch angle in appropriate quadrant and show all work.

Answer 1

Given:

The angle is 240 degrees.

Required:

We need to find the six trigonometric functions.

Step-by-step explanation:

Consider the sine function.

`sin(240\degree)=sin(180+60)\degree`

`=-sin(60)\degree`
`=(-√(3))/(2)`

`sin(240\degree)=(-√(3))/(2)`

Consider the cosine function.

`cos(240\degree)=cos(180+60)\degree`

`=-cos(60)\degree`
`=-(1)/(2)`

`cos(240\degree)=-(1)/(2)`

Consider the tan function.

`tan(240)\degree=(sin(240)\degree)/(cos(240)\degree)`

`Substitute\text{ }sin(240\degree)=(-√(3))/(2)\text{ and }cos(240\degree)=-\frac{1}{2\text{ }}\text{ in the equation.}`

`tan(240)\degree=(-(√(3))/(2))/(-(1)/(2))`

`tan(240)\degree=-(√(3))/(2)*(-(2)/(1))`

`tan(240)\degree=√(3)`

Consider the cot function.

`cot(240)\degree=(cos(240)\degree)/(sin(240)\degree)`

`Substitute\text{ }sin(240\degree)=(-√(3))/(2)\text{ and }cos(240\degree)=-\frac{1}{2\text{ }}\text{ in the equation.}`

`cot(240)\degree=(-(1)/(2))/(-(√(3))/(2))`

`cot(240)\degree=-(1)/(2)*(-(2)/(√(3)))`

`cot(240)\degree=(1)/(√(3))`

Consider the sec function.

`sec(240)\degree=(1)/(cos(240)\degree)`

`Substitute\text{ }cos(240\degree)=-\frac{1}{2\text{ }}\text{ in the equation.}`

`sec(240)\degree=(1)/(-(1)/(2))`

`sec(240)\degree=1*(-(2)/(1))`

`sec(240)\degree=-2`

Consider the csc function.

`csc(240)\degree=(1)/(sin(240)\degree)`

`Substitute\text{ }sin(240\degree)=(-√(3))/(2)\text{ in the equation.}`

`csc(240)\degree=(1)/(-(√(3))/(2))`

`csc(240)\degree=1*((-2)/(√(3)))`

`csc(240)\degree=(-2)/(√(3))`

Consider the angle in a graph:

Final answer:

`sin(240\degree)=(-√(3))/(2)`

`cos(240\degree)=-(1)/(2)`
`tan(240)\degree=√(3)`

`cot(240)\degree=(1)/(√(3))`

`sec(240)\degree=-2`

`csc(240)\degree=(-2)/(√(3))`

The angle lies in the third quadrant.