Find the Domain 2cot(3x)^2=7csc(3x-5)

Problem

2*cot2(3*x)=7*csc(3*x−5)

Solution

1. Identify the trigonometric functions involved in the equation. The function cot(3*x) is defined as cos(3*x)/sin(3*x) and the function csc(3*x−5) is defined as 1/sin(3*x−5)

2. Determine the restrictions for the cotangent term. The expression cot(3*x) is undefined when its denominator is zero:

sin(3*x)=0

3. Solve for x in the first restriction. The sine function is zero at integer multiples of π

3*x=n*π

x=(n*π)/3

where n is any integer.

4. Determine the restrictions for the cosecant term. The expression csc(3*x−5) is undefined when its denominator is zero:

sin(3*x−5)=0

5. Solve for x in the second restriction. Set the argument equal to integer multiples of π

3*x−5=k*π

3*x=k*π+5

x=(k*π+5)/3

where k is any integer.

6. Combine the restrictions to define the domain. The domain consists of all real numbers x except for the values found in steps 3 and 5.

Final Answer

Domain={[x∈ℝ,x≠(n*π)/3,x≠(k*π+5)/3,n,k∈ℤ]}

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