The cosecant (csc) of an angle is the reciprocal of the sine of that angle. In other words, csc θ = 1/sin θ.

### Key Facts

- Definition of csc: The cosecant (csc) of an angle is the reciprocal of the sine of that angle. In other words, csc θ = 1/sin θ.
- Conversion to degrees: To make the calculations easier, we can convert the angle from radians to degrees. Using the conversion factor 180/π, we have (5π/3) × (180/π) = 300°.
- Reference angle: The reference angle for 300° is 60°. In the fourth quadrant, the sine function is negative.
- Value of sine: The sine of 60° is √3/2.

Based on these facts, we can now calculate the exact value of csc (5π/3):

csc (5π/3) = 1/sin (5π/3)

= 1/sin (2π – π/3)

= 1/[sin (2π)cos (π/3) – cos (2π)sin (π/3)]

= 1/[0 – 1(√3/2)]

= 1/(-√3/2)

= -2/√3

= -(2√3)/3

Therefore, the exact value of csc (5π/3) is -(2√3)/3.

## Solution

To find the exact value of csc (5π/3), we can follow these steps:

### Step 1: Convert to Degrees

To make the calculations easier, we can convert the angle from radians to degrees. Using the conversion factor 180/π, we have (5π/3) × (180/π) = 300°.

### Step 2: Find the Reference Angle

The reference angle for 300° is 60°. In the fourth quadrant, the sine function is negative.

### Step 3: Find the Value of Sine

The sine of 60° is √3/2.

### Step 4: Calculate the Value of CSC

Now we can calculate the exact value of csc (5π/3):

csc (5π/3) = 1/sin (5π/3)

= 1/sin (2π – π/3)

= 1/[sin (2π)cos (π/3) – cos (2π)sin (π/3)]

= 1/[0 – 1(√3/2)]

= 1/(-√3/2)

= -2/√3

= -(2√3)/3

### Conclusion

Therefore, the exact value of csc (5π/3) is -(2√3)/3.

## Sources

- https://socratic.org/questions/how-do-you-find-the-value-of-csc-5pi-3
- https://www.wyzant.com/resources/answers/172593/how_do_you_find_csc_of_5pi_3
- https://socratic.org/questions/how-do-you-evaluate-csc-5pi-3

## FAQs

### What is the definition of cosecant?

The cosecant (csc) of an angle is the reciprocal of the sine of that angle. In other words, csc θ = 1/sin θ.

### How do I convert radians to degrees?

To convert radians to degrees, multiply the radian measure by 180/π.

### What is the reference angle for an angle in the fourth quadrant?

To find the reference angle for an angle in the fourth quadrant, subtract the angle from 360 degrees.

### What is the sine of 60 degrees?

The sine of 60 degrees is √3/2.

### How do I calculate the exact value of csc (5π/3)?

To calculate the exact value of csc (5π/3), follow these steps:

- Convert (5π/3) to degrees: (5π/3) × (180/π) = 300°
- Find the reference angle for 300°: 360° – 300° = 60°
- Find the sine of 60°: sin 60° = √3/2
- Calculate csc (5π/3): csc (5π/3) = 1/sin (5π/3) = 1/(√3/2) = -2/√3 = -(2√3)/3

### Can I use a calculator to find the exact value of csc (5π/3)?

Yes, you can use a calculator to find the exact value of csc (5π/3). However, it is important to make sure that your calculator is in radian mode.

### What is the domain of the cosecant function?

The domain of the cosecant function is all real numbers except for the values where the sine function is equal to zero.

### What is the range of the cosecant function?

The range of the cosecant function is all real numbers except for the values between -1 and 1.