Find the Domain sin(x) = square root of 3/2

Problem

sin(x)=√(,3)/2

Solution

1. Identify the nature of the task. The equation sin(x)=√(,3)/2 is a trigonometric equation to be solved for x In this context, "Find the Domain" refers to finding the set of all possible values of x that satisfy the equation.

2. Determine the reference angle. We look for the angle in the first quadrant where the sine value is √(,3)/2

sin(π/3)=√(,3)/2

3. Find the second solution within one period [0,2*π) Since the sine function is positive in the second quadrant, we use the identity sin(π−θ)=sin(θ)

x=π−π/3=(2*π)/3

4. Generalize the solution to include all possible values. Because the sine function has a period of 2*π we add 2*π*k to each solution, where k is any integer.

x=π/3+2*π*k

x=(2*π)/3+2*π*k

Final Answer

sin(x)=√(,3)/2⇒x=π/3+2*π*k,(2*π)/3+2*π*k* for *k∈ℤ

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