What is the general solution for sin(x+pi/3)= 1/2 cos(x-pi/6) ?

The general solution for sin(x+pi/3)= 1/2 cos(x-pi/6) is x=(2pi)/3+pin

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sin(x+pi/3)= 1/2 cos(x-pi/6)

sin (x + \frac{π}{3}) = \frac{1}{2} cos (x - \frac{π}{6})

x = \frac{2 π}{3} + πn

Solution steps

sin (x + \frac{π}{3}) = \frac{1}{2} cos (x - \frac{π}{6})

Rewrite using trig identities

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

Subtract

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

from both sides

\frac{1}{4} sin (x) + \frac{3}{4} cos (x) = 0

Simplify

\frac{1}{4} sin (x) + \frac{3}{4} cos (x) : \frac{sin ( x ) + 3 cos ( x )}{4}

\frac{sin ( x ) + 3 cos ( x )}{4} = 0

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

sin (x) + 3 cos (x) = 0

Rewrite using trig identities

tan (x) + 3 = 0

Move

3

to the right side

tan (x) = - 3

General solutions for

tan (x) = - 3

x = \frac{2 π}{3} + πn

\frac{2 sec ^{2} ( x ) - 1}{sec ^{2} ( x )} = sec^{2} (x)

9 (sin (x) - 0.6 π) + 8 = 0

t, v = 100 cos (50 t + 2 0^{\circ}) v

cos (2 x) = sin (\frac{π}{2} - x)

sin (x) = sin (5 0^{\circ})

What is the general solution for sin(x+pi/3)= 1/2 cos(x-pi/6) ?
The general solution for sin(x+pi/3)= 1/2 cos(x-pi/6) is x=(2pi)/3+pin