Use the unit circle to find the sin(240) and cos(240), without using a calculator.

Question

please help me get the answer to this question please.

Answer 1

We can use the addition/subtraction angle identities.  To do this, pick two angles that we can easily evaluate sine and cosine of in a unit circle.  These angles must add up to or make a difference of the original angle.

 

sin(240) = sin(270 - 30)

 

 

sin(240) = sin(180 + 60)             works too.

 

 

Use the subtraction angle identity for sine to find the value of sin(240).  30, 60, 180 and 270 are obvious angles on a unit circle.

 

sin(240) = sin(270 - 30)

 

             = sin(270)cos(30) - cos(270)sin(30)

 

 

        or

 

Addition angle identity for sine.

 

 

sin(240) = sin(180 + 60)

 

             = sin(180)cos(60) + cos(180)sin(60)           works too.

 

 

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Subtraction angle identity for cosine.

 

 

cos(240) = cos(270 - 30)

 

              = cos(270)cos(30) + sin(270)sin(30)

 

 

               or

 

Addition angle identity for cosine.

 

 

cos(240) = cos(180 + 60)

 

              = cos(180)cos(60) - sin(180)sin(60)         works too.

Answer 2

The reference angle for 240° is 60°.

Determine the sine and cosine for 60° and adjust the sign for QIII.