How do you find csc of 5pi/3?

Question

This is a trigonometry unit circle problem. 

 

Answer 1

Hi Jocelyn,

 

 

Using our unit circle, we can figure that 5π / 3 is roughly equal to 300º. Now, if we look a little closer at our unit circle, we can get our cos and sin values from it.

 

 

• Remember that on the unit circle, the coordinate points are read as (cos, sin) values. The first value in the parentheses are cos and the second value in the parentheses are sin.

 

 

• For 5π / 3, we get the ordered pair ( 1/2, -√3/2). This means that our cos value for 5π / 3 will be 1/2 and our sin value will be -√3/2, which you can also rewrite the following way:

 

cos (5π / 3) = 1/2

 

sin (5π / 3) = -√3/2

 

 

• csc is the same thing as 1 / sin, so you can just apply that to your problem here as such:

 

 

csc (5π / 3) = 1 / [sin (5π / 3)] = 1 / (-√3/2) = -2/√3

 

 

 

 

Source: http://etc.usf.edu/clipart/43200/43215/unit-circle7_43215_sm.gif

Answer 2

First, we can write the given angle as a sum or difference of angles that we can easily identify sine and cosine of.

 

5π / 3 = (6π - π) / 3 = 2π - (π/3)

 

2π and π/3 are angles we can easily identify sine and cosine of. 

 

 

csc(5π / 3) = 1 / [sin(5π / 3)]

 

                 = 1 / [sin(2π - π/3)]

 

 

We can use the subtraction angle identity for sine.

 

csc(5π / 3) = 1 / [sin(2π)cos(π/3) - cos(2π)sin(π/3)]

 

                 = 1 / [0 - 1(√(3) / 2))]

 

                 = 1 / (-√(3) / 2)

 

                 = - 2 / √(3)

 

 

Rationalize the denominator.

 

csc(5π / 3) = - (2/3)√3

Answer 3

csc of 5pi/3

5pi/3=5*180/3=300º

300º is in the fourth quadrant, where sin is negative

csc=1/sin

sin(300º)=(-√3/2)

csc(300º)=1/(-√3/2)=(-2/√3)