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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Popular Trigonometry >

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

Solution

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

Solution

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

Degrees

x = 70.52877 \dots^{\circ} + 36 0^{\circ} n, x = 289.47122 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

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

Factor out common term

cos (x)

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

\frac{3}{2} + \frac{3}{2} = 3

\frac{3}{2} + \frac{3}{2}

Apply rule

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

= \frac{3 + 3}{2}

Factor

3 + 3 : 23

3 + 3

Factor out common term

3

= 3 (1 + 1)

Refine

= 23

= \frac{2 3}{2}

Divide the numbers:

\frac{2}{2} = 1

= 3

= 3 cos (x)

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

Add similar elements:

\frac{1}{2} sin (x) - \frac{1}{2} sin (x) = 0

\frac{1}{2} sin (x) - \frac{1}{2} sin (x)

Factor out common term

sin (x)

= sin (x) (\frac{1}{2} - \frac{1}{2})

\frac{1}{2} - \frac{1}{2} = 0

\frac{1}{2} - \frac{1}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{1 - 1}{2}

Refine

= 0

= 0

= 3 cos (x)

\frac{3}{3} = \frac{1}{3}

\frac{3}{3}

Apply radical rule:

3 = 3^{\frac{1}{2}}

= \frac{3 ^{\frac{1}{2}}}{3}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{3 ^{\frac{1}{2}}}{3 ^{1}} = \frac{1}{3 ^{1 - \frac{1}{2}}}

= \frac{1}{3 ^{1 - \frac{1}{2}}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{1}{3 ^{\frac{1}{2}}}

Apply radical rule:

3^{\frac{1}{2}} = 3

= \frac{1}{3}

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

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}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

3 cos (x) 3 - 1 = 3 cos (x) - 1

3 cos (x) 3 - 1

Apply radical rule:

a a = a

33 = 3

= 3 cos (x) - 1

= \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 = 0

Add

1

to both sides

3 cos (x) - 1 + 1 = 0 + 1

Simplify

3 cos (x) = 1

3 cos (x) = 1

Divide both sides by

3

3 cos (x) = 1

Divide both sides by

3

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

Simplify

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

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

Apply trig inverse properties

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

General solutions for

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

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

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

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

Graph

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Frequently Asked Questions (FAQ)

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