Solve for x sin(x)=5/13

Problem

sin(x)=5/13

Solution

1. Identify the equation as a basic trigonometric equation where the sine of an angle x is a positive constant.

2. Apply the inverse sine function to find the principal value (the reference angle) in the first quadrant.

x=arcsin(5/13)

3. Determine the general solution for the first quadrant by adding multiples of the period 2*π

x=arcsin(5/13)+2*π*n

4. Determine the second set of solutions using the identity sin(π−θ)=sin(θ) which places the angle in the second quadrant.

x=π−arcsin(5/13)+2*π*n

5. Specify the domain for the integer n to complete the general solution.

n∈ℤ

Final Answer

sin(x)=5/13⇒x=arcsin(5/13)+2*π*n,π−arcsin(5/13)+2*π*n

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