What is the general solution for sin(x+pi/3)+cos(x+pi/6)=(sqrt(3))/3 ?

The general solution for sin(x+pi/3)+cos(x+pi/6)=(sqrt(3))/3 is x=1.23095…+2pin,x=2pi-1.23095…+2pin

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sin(x+pi/3)+cos(x+pi/6)=(sqrt(3))/3

sin (x + \frac{π}{3}) + cos (x + \frac{π}{6}) = \frac{3}{3}

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

Solution steps

sin (x + \frac{π}{3}) + cos (x + \frac{π}{6}) = \frac{3}{3}

Rewrite using trig identities

3 cos (x) = \frac{1}{3}

Subtract

\frac{1}{3}

from both sides

3 cos (x) - \frac{1}{3} = 0

Simplify

3 cos (x) - \frac{1}{3} : \frac{3 cos ( x ) - 1}{3}

\frac{3 cos ( x ) - 1}{3} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 cos (x) - 1 = 0

Move

1

to the right side

3 cos (x) = 1

Divide both sides by

3

cos (x) = \frac{1}{3}

Apply trig inverse properties

x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

Show solutions in decimal form

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

4 sin^{2} (x) cot (x) - 3 cot (x) = 0

sin (2 x) = - 2 cos^{2} (x)

cos (\frac{2 x + 3}{2}) = - \frac{1}{2}

1 = cot^{2} (x) + csc (x)

sin (\frac{θ}{2}) = \frac{3}{2}

What is the general solution for sin(x+pi/3)+cos(x+pi/6)=(sqrt(3))/3 ?
The general solution for sin(x+pi/3)+cos(x+pi/6)=(sqrt(3))/3 is x=1.23095…+2pin,x=2pi-1.23095…+2pin