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

Problem

3*sin(x)−3*cos(x)=1

Solution

1. Divide both sides of the equation by 3 to isolate the trigonometric terms.

sin(x)−cos(x)=1/3

2. Apply the harmonic identity a*sin(x)+b*cos(x)=R*sin(x+α) where R=√(,a2+b2) and tan(α)=b/a

R=√(,1+(−1)2)=√(,2)

tan(α)=(−1)/1⇒α=−π/4

3. Rewrite the equation using the identity.

√(,2)*sin(x−π/4)=1/3

4. Isolate the sine function by dividing by √(,2)

sin(x−π/4)=1/(3√(,2))

5. Solve for the argument using the inverse sine function, accounting for the general solution of sine.

x−π/4=arcsin(1/(3√(,2)))+2*k*π

x−π/4=π−arcsin(1/(3√(,2)))+2*k*π

6. Isolate x by adding π/4 to both sides for both cases, where k is any integer.

x=π/4+arcsin(1/(3√(,2)))+2*k*π

x=(5*π)/4−arcsin(1/(3√(,2)))+2*k*π

Final Answer

x=π/4+arcsin(√(,2)/6)+2*k*π,(5*π)/4−arcsin(√(,2)/6)+2*k*π

Want more problems? Check here!