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)

Solution

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

Solution

x = \frac{2 π}{3} + πn

Degrees

x = 12 0^{\circ} + 18 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Rewrite using trig identities

sin (x + \frac{π}{3})

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

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

Simplify

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

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

Simplify

cos (\frac{π}{3}) : \frac{1}{2}

cos (\frac{π}{3})

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{1}{2}

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

Simplify

sin (\frac{π}{3}) : \frac{3}{2}

sin (\frac{π}{3})

Use the following trivial identity:

sin (\frac{π}{3}) = \frac{3}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

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

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

Use the Angle Difference identity:

cos (s - t) = cos (s) cos (t) + sin (s) sin (t)

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

Simplify

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

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

Simplify

cos (\frac{π}{6}) : \frac{3}{2}

cos (\frac{π}{6})

Use the following trivial identity:

cos (\frac{π}{6}) = \frac{3}{2}

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

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

Simplify

sin (\frac{π}{6}) : \frac{1}{2}

sin (\frac{π}{6})

Use the following trivial identity:

sin (\frac{π}{6}) = \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

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

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

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

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

\frac{1}{4} sin (x) = \frac{sin ( x )}{4}

\frac{1}{4} sin (x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot sin ( x )}{4}

Multiply:

1 \cdot sin (x) = sin (x)

= \frac{sin ( x )}{4}

\frac{3}{4} cos (x) = \frac{3 cos ( x )}{4}

\frac{3}{4} cos (x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 cos ( x )}{4}

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

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \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

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

Divide both sides by

cos (x), cos (x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

tan (x) + 3 = 0

tan (x) + 3 = 0

Move

3

to the right side

tan (x) + 3 = 0

Subtract

3

from both sides

tan (x) + 3 - 3 = 0 - 3

Simplify

tan (x) = - 3

tan (x) = - 3

General solutions for

tan (x) = - 3

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

Popular Examples

(2sec^2(x)-1)/(sec^2(x))=sec^2(x)

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

9(sin(x)-0.6pi)+8=0

9 (sin (x) - 0.6 π) + 8 = 0

solvefor t,v=100cos(50t+20)v solve for

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

cos(2x)=sin(pi/2-x)

cos (2 x) = sin (\frac{π}{2} - x)

sin(x)=sin(50)

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

Frequently Asked Questions (FAQ)

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