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

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

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sin(x+pi/6)+cos(x+pi/6)=1

Solution

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

Solution

x = 2 πn - \frac{π}{6}, x = 2 πn + \frac{π}{3}

Degrees

x = - 3 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

sin (\frac{π}{6} + x) + cos (\frac{π}{6} + x) = 2 sin (\frac{π}{6} + x + \frac{π}{4})

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

Rewrite as

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

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{1}{2}

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (\frac{π}{4}) sin (\frac{π}{6} + x) + sin (\frac{π}{4}) cos (\frac{π}{6} + x))

Use the Angle Sum identity:

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

= 2 sin (\frac{π}{6} + x + \frac{π}{4})

= 2 sin (\frac{π}{6} + x + \frac{π}{4})

2 sin (\frac{π}{6} + x + \frac{π}{4}) = 1

Divide both sides by

2

2 sin (\frac{π}{6} + x + \frac{π}{4}) = 1

Divide both sides by

2

\frac{2 sin ( \frac{π}{6} + x + \frac{π}{4} )}{2} = \frac{1}{2}

Simplify

\frac{2 sin ( \frac{π}{6} + x + \frac{π}{4} )}{2} = \frac{1}{2}

Simplify

\frac{2 sin ( \frac{π}{6} + x + \frac{π}{4} )}{2} : sin (\frac{π}{6} + x + \frac{π}{4})

\frac{2 sin ( \frac{π}{6} + x + \frac{π}{4} )}{2}

Cancel the common factor:

2

= sin (\frac{π}{6} + x + \frac{π}{4})

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

sin (\frac{π}{6} + x + \frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{6} + x + \frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{6} + x + \frac{π}{4}) = \frac{2}{2}

General solutions for

sin (\frac{π}{6} + x + \frac{π}{4}) = \frac{2}{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{π}{6} + x + \frac{π}{4} = \frac{π}{4} + 2 πn, \frac{π}{6} + x + \frac{π}{4} = \frac{3 π}{4} + 2 πn

\frac{π}{6} + x + \frac{π}{4} = \frac{π}{4} + 2 πn, \frac{π}{6} + x + \frac{π}{4} = \frac{3 π}{4} + 2 πn

Solve

\frac{π}{6} + x + \frac{π}{4} = \frac{π}{4} + 2 πn : x = 2 πn - \frac{π}{6}

\frac{π}{6} + x + \frac{π}{4} = \frac{π}{4} + 2 πn

Subtract

\frac{π}{4}

from both sides

\frac{π}{6} + x + \frac{π}{4} - \frac{π}{4} = \frac{π}{4} + 2 πn - \frac{π}{4}

Simplify

\frac{π}{6} + x = 2 πn

Move

\frac{π}{6}

to the right side

\frac{π}{6} + x = 2 πn

Subtract

\frac{π}{6}

from both sides

\frac{π}{6} + x - \frac{π}{6} = 2 πn - \frac{π}{6}

Simplify

x = 2 πn - \frac{π}{6}

x = 2 πn - \frac{π}{6}

Solve

\frac{π}{6} + x + \frac{π}{4} = \frac{3 π}{4} + 2 πn : x = 2 πn + \frac{π}{3}

\frac{π}{6} + x + \frac{π}{4} = \frac{3 π}{4} + 2 πn

Move

\frac{π}{6}

to the right side

\frac{π}{6} + x + \frac{π}{4} = \frac{3 π}{4} + 2 πn

Subtract

\frac{π}{6}

from both sides

\frac{π}{6} + x + \frac{π}{4} - \frac{π}{6} = \frac{3 π}{4} + 2 πn - \frac{π}{6}

Simplify

\frac{π}{6} + x + \frac{π}{4} - \frac{π}{6} = \frac{3 π}{4} + 2 πn - \frac{π}{6}

Simplify

\frac{π}{6} + x + \frac{π}{4} - \frac{π}{6} : x + \frac{π}{4}

\frac{π}{6} + x + \frac{π}{4} - \frac{π}{6}

Add similar elements:

\frac{π}{6} - \frac{π}{6} = 0

= x + \frac{π}{4}

Simplify

\frac{3 π}{4} + 2 πn - \frac{π}{6} : 2 πn + \frac{7 π}{12}

\frac{3 π}{4} + 2 πn - \frac{π}{6}

Group like terms

= 2 πn - \frac{π}{6} + \frac{3 π}{4}

Least Common Multiplier of

6, 4 : 12

6, 4

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

6

4

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{6} :

multiply the denominator and numerator by

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

For

\frac{3 π}{4} :

multiply the denominator and numerator by

3

\frac{3 π}{4} = \frac{3 π 3}{4 \cdot 3} = \frac{9 π}{12}

= - \frac{π 2}{12} + \frac{9 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 2 + 9 π}{12}

Add similar elements:

- 2 π + 9 π = 7 π

= 2 πn + \frac{7 π}{12}

x + \frac{π}{4} = 2 πn + \frac{7 π}{12}

x + \frac{π}{4} = 2 πn + \frac{7 π}{12}

x + \frac{π}{4} = 2 πn + \frac{7 π}{12}

Move

\frac{π}{4}

to the right side

x + \frac{π}{4} = 2 πn + \frac{7 π}{12}

Subtract

\frac{π}{4}

from both sides

x + \frac{π}{4} - \frac{π}{4} = 2 πn + \frac{7 π}{12} - \frac{π}{4}

Simplify

x + \frac{π}{4} - \frac{π}{4} = 2 πn + \frac{7 π}{12} - \frac{π}{4}

Simplify

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= x

Simplify

2 πn + \frac{7 π}{12} - \frac{π}{4} : 2 πn + \frac{π}{3}

2 πn + \frac{7 π}{12} - \frac{π}{4}

Least Common Multiplier of

12, 4 : 12

12, 4

Least Common Multiplier (LCM)

Prime factorization of

12 : 2 \cdot 2 \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

12

4

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

= \frac{7 π}{12} - \frac{π 3}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{7 π - π 3}{12}

Add similar elements:

7 π - 3 π = 4 π

= \frac{4 π}{12}

Cancel the common factor:

4

= 2 πn + \frac{π}{3}

x = 2 πn + \frac{π}{3}

x = 2 πn + \frac{π}{3}

x = 2 πn + \frac{π}{3}

x = 2 πn - \frac{π}{6}, x = 2 πn + \frac{π}{3}

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Frequently Asked Questions (FAQ)

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