Solve for x 2cos(x)+2sin(x) = square root of 6

Problem

2*cos(x)+2*sin(x)=√(,6)

Solution

1. Divide both sides of the equation by 2 to isolate the trigonometric terms.

cos(x)+sin(x)=√(,6)/2

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

R=√(,1+1)=√(,2)

tan(α)=1/1⇒α=π/4

3. Rewrite the equation using the identity.

√(,2)*cos(x−π/4)=√(,6)/2

4. Isolate the cosine term by dividing both sides by √(,2)

cos(x−π/4)=√(,6)/(2√(,2))

cos(x−π/4)=√(,3)/2

5. Solve for the argument x−π/4 using the general solution for cosine.

x−π/4=±π/6+2*k*π

6. Isolate x by adding π/4 to both cases.

x=π/4+π/6+2*k*π⇒x=(5*π)/12+2*k*π

x=π/4−π/6+2*k*π⇒x=π/12+2*k*π

Final Answer

x=π/12+2*k*π,(5*π)/12+2*k*π

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