Solve for x sin(x)=-3/5

Problem

sin(x)=−3/5

Solution

1. Identify the range of the sine function. Since −1≤−3/5≤1 solutions exist for x

2. Calculate the reference angle (x_ref) by taking the inverse sine of the absolute value of the ratio.

(x_ref)=arcsin(3/5)

3. Determine the quadrants where the sine function is negative. Sine is negative in Quadrant III and Quadrant IV.

4. Find the general solution for Quadrant III.

x=π+arcsin(3/5)+2*π*n

5. Find the general solution for Quadrant IV.

x=2*π−arcsin(3/5)+2*π*n

6. Express the solution using the standard inverse sine notation, where n is any integer.

x=arcsin(−3/5)+2*π*n

x=π−arcsin(−3/5)+2*π*n

Final Answer

sin(x)=−3/5⇒x=(−1)n*arcsin(−3/5)+n*π

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