Determine the exact value for the following trigonometric ratios.

Question

1.sin(5Π/3)

2.cos(5π/3)

3.cot(5Π/3)

Answer 1

 

(1) sin 5pi/3

 

    pi/3 = 5(180)/3 = 300 degrees

    This produces a 60 degree reference angle in quadrant 4.

    Sine = the y-value, which is -√3/2

 

(2) cos 5pi/3

 

    This is the same angle as in number 1.

    Cos = the x-value, which is 1/2

 

(3) cot 5pi/3

 

     Again, same angle

     cotangent is the reciprocal of tangent.

     tangent is sine/cosine ... cotangent is cosine/sine

     cotangent = cosine/sine

                    = (1/2)/(√3/2)

                    = (1/2)•(2/√3)

                    = 2/(2√3)

                    = 1/√3

                    = √3/3

 

Answer 2

Let find the exact value using the addition/subtraction angle identities for sine, cosine and tangent.

 

To do this, we think of 2 angles in which sine and cosine of is obvious to us on a unit circle.  Those two angles must add up to the angle given.

 

sin(5π / 3) = sin(6π/3 - π/3)

 

                = sin(2π - π/3)

 

 

The two angles being subtracted are angles on the unit circle.  Thus, we can use the subtraction angle identity for sine.

 

sin(5π / 3) = sin(2π)cos(π/3) - cos(2π)sin(π/3)

 

                = -1 * √(3)/2

 

                = -√(3)/2

 

 

sin(5π / 3) = -√(3)/2

 

 

Next, to find cos(5π/3), can use the identity               sin²(5π/3)+cos²(5π/3)=1

 

 

Plug in the value of sin(5π/3) we just found into this identity to get cos(5π/3).   Since this angle is in the 4th quadrant, cos(5π/3) will be positive.

 

 

Then once you have your cos(5π/3), use the identity

 

 

cot(5π/3) = cos(5π/3) / sin(5π/3)

 

 

to find cot(5π/3).