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.