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

User Antonversal
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1 Answer

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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.

What are the 6 trig functions of a 240 degree angle. Sketch angle in appropriate quadrant-example-1
What are the 6 trig functions of a 240 degree angle. Sketch angle in appropriate quadrant-example-2
User Tom Tresansky
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