Solve for x 3sin(x)-2cos(x)=1

Problem

3*sin(x)−2*cos(x)=1

Solution

1. Express the equation in the form R*sin(x−α)=1 by using the harmonic addition theorem, where R=√(,3+(−2)2)

R=√(,9+4)

R=√(,13)

2. Determine the phase shift α by solving tan(α)=2/3

α=arctan(2/3)

3. Rewrite the original equation using the calculated values.

√(,13)*sin(x−α)=1

4. Isolate the sine function by dividing both sides by √(,13)

sin(x−α)=1/√(,13)

5. Solve for the general angle x−α using the inverse sine function.

x−α=arcsin(1/√(,13))+2*n*π

x−α=π−arcsin(1/√(,13))+2*n*π

6. Isolate x by adding α to both sides of the general solutions.

x=arctan(2/3)+arcsin(1/√(,13))+2*n*π

x=arctan(2/3)+π−arcsin(1/√(,13))+2*n*π

Final Answer

x=arctan(2/3)+(−1)n*arcsin(1/√(,13))+n*π

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