Find the Domain cos(2z)+sin(2z)=2

Problem

cos(2*z)+sin(2*z)=2

Solution

1. Identify the range of the trigonometric functions. For any real number θ the functions cos(θ) and sin(θ) are bounded such that −1≤cos(θ)≤1 and −1≤sin(θ)≤1

2. Analyze the maximum possible value of the sum. The maximum value of the expression cos(2*z)+sin(2*z) can be found using the identity a*cos(θ)+b*sin(θ)=√(,a2+b2)*cos(θ−ϕ)

3. Calculate the maximum value. For 1⋅cos(2*z)+1⋅sin(2*z) the maximum value is √(,1+1)=√(,2)

4. Compare the maximum value to the given equation. Since √(,2)≈1.414 the maximum value of the left-hand side is less than 2

5. Determine the solution set. Because cos(2*z)+sin(2*z) can never reach the value 2 for any real z there are no real solutions.

6. Conclude the domain. The "domain" in the context of an equation usually refers to the set of values for which the expression is defined. Both cos(2*z) and sin(2*z) are defined for all real numbers.

Final Answer

Domain=(−∞,∞)

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