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sin(3x)+cos(3x)=-(sqrt(2))/2

Solution

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

Solution

x = \frac{2 πn}{3} + \frac{11 π}{36}, x = \frac{2 πn}{3} + \frac{19 π}{36}

Degrees

x = 5 5^{\circ} + 12 0^{\circ} n, x = 9 5^{\circ} + 12 0^{\circ} n

Solution steps

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

Rewrite using trig identities

sin (3 x) + cos (3 x)

sin (3 x) + cos (3 x) = 2 sin (3 x + \frac{π}{4})

sin (3 x) + cos (3 x)

Rewrite as

= 2 (\frac{1}{2} sin (3 x) + \frac{1}{2} cos (3 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 (3 x) + sin (\frac{π}{4}) cos (3 x))

Use the Angle Sum identity:

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

= 2 sin (3 x + \frac{π}{4})

= 2 sin (3 x + \frac{π}{4})

2 sin (3 x + \frac{π}{4}) = - \frac{2}{2}

Divide both sides by

2

2 sin (3 x + \frac{π}{4}) = - \frac{2}{2}

Divide both sides by

2

\frac{2 sin ( 3 x + \frac{π}{4} )}{2} = \frac{- \frac{2}{2}}{2}

Simplify

\frac{2 sin ( 3 x + \frac{π}{4} )}{2} = \frac{- \frac{2}{2}}{2}

Simplify

\frac{2 sin ( 3 x + \frac{π}{4} )}{2} : sin (3 x + \frac{π}{4})

\frac{2 sin ( 3 x + \frac{π}{4} )}{2}

Cancel the common factor:

2

= sin (3 x + \frac{π}{4})

Simplify

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

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

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

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

Apply the fraction rule:

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

\frac{\frac{2}{2}}{2} = \frac{2}{2 2}

= - \frac{2}{2 2}

Rationalize

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

- \frac{2}{2 2}

Multiply by the conjugate

\frac{2}{2}

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

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

222 = 4

222

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

= - \frac{1}{2}

sin (3 x + \frac{π}{4}) = - \frac{1}{2}

sin (3 x + \frac{π}{4}) = - \frac{1}{2}

sin (3 x + \frac{π}{4}) = - \frac{1}{2}

General solutions for

sin (3 x + \frac{π}{4}) = - \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}

3 x + \frac{π}{4} = \frac{7 π}{6} + 2 πn, 3 x + \frac{π}{4} = \frac{11 π}{6} + 2 πn

3 x + \frac{π}{4} = \frac{7 π}{6} + 2 πn, 3 x + \frac{π}{4} = \frac{11 π}{6} + 2 πn

Solve

3 x + \frac{π}{4} = \frac{7 π}{6} + 2 πn : x = \frac{2 πn}{3} + \frac{11 π}{36}

3 x + \frac{π}{4} = \frac{7 π}{6} + 2 πn

Move

\frac{π}{4}

to the right side

3 x + \frac{π}{4} = \frac{7 π}{6} + 2 πn

Subtract

\frac{π}{4}

from both sides

3 x + \frac{π}{4} - \frac{π}{4} = \frac{7 π}{6} + 2 πn - \frac{π}{4}

Simplify

3 x + \frac{π}{4} - \frac{π}{4} = \frac{7 π}{6} + 2 πn - \frac{π}{4}

Simplify

3 x + \frac{π}{4} - \frac{π}{4} : 3 x

3 x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= 3 x

Simplify

\frac{7 π}{6} + 2 πn - \frac{π}{4} : 2 πn + \frac{11 π}{12}

\frac{7 π}{6} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{4} + \frac{7 π}{6}

Least Common Multiplier of

4, 6 : 12

4, 6

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

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

4

6

= 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}

For

\frac{7 π}{6} :

multiply the denominator and numerator by

2

\frac{7 π}{6} = \frac{7 π 2}{6 \cdot 2} = \frac{14 π}{12}

= - \frac{π 3}{12} + \frac{14 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 3 + 14 π}{12}

Add similar elements:

- 3 π + 14 π = 11 π

= 2 πn + \frac{11 π}{12}

3 x = 2 πn + \frac{11 π}{12}

3 x = 2 πn + \frac{11 π}{12}

3 x = 2 πn + \frac{11 π}{12}

Divide both sides by

3

3 x = 2 πn + \frac{11 π}{12}

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{11 π}{12}}{3}

Simplify

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{11 π}{12}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{2 πn}{3} + \frac{\frac{11 π}{12}}{3} : \frac{2 πn}{3} + \frac{11 π}{36}

\frac{2 πn}{3} + \frac{\frac{11 π}{12}}{3}

\frac{\frac{11 π}{12}}{3} = \frac{11 π}{36}

\frac{\frac{11 π}{12}}{3}

Apply the fraction rule:

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

= \frac{11 π}{12 \cdot 3}

Multiply the numbers:

12 \cdot 3 = 36

= \frac{11 π}{36}

= \frac{2 πn}{3} + \frac{11 π}{36}

x = \frac{2 πn}{3} + \frac{11 π}{36}

x = \frac{2 πn}{3} + \frac{11 π}{36}

x = \frac{2 πn}{3} + \frac{11 π}{36}

Solve

3 x + \frac{π}{4} = \frac{11 π}{6} + 2 πn : x = \frac{2 πn}{3} + \frac{19 π}{36}

3 x + \frac{π}{4} = \frac{11 π}{6} + 2 πn

Move

\frac{π}{4}

to the right side

3 x + \frac{π}{4} = \frac{11 π}{6} + 2 πn

Subtract

\frac{π}{4}

from both sides

3 x + \frac{π}{4} - \frac{π}{4} = \frac{11 π}{6} + 2 πn - \frac{π}{4}

Simplify

3 x + \frac{π}{4} - \frac{π}{4} = \frac{11 π}{6} + 2 πn - \frac{π}{4}

Simplify

3 x + \frac{π}{4} - \frac{π}{4} : 3 x

3 x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= 3 x

Simplify

\frac{11 π}{6} + 2 πn - \frac{π}{4} : 2 πn + \frac{19 π}{12}

\frac{11 π}{6} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{4} + \frac{11 π}{6}

Least Common Multiplier of

4, 6 : 12

4, 6

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

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

4

6

= 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}

For

\frac{11 π}{6} :

multiply the denominator and numerator by

2

\frac{11 π}{6} = \frac{11 π 2}{6 \cdot 2} = \frac{22 π}{12}

= - \frac{π 3}{12} + \frac{22 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 3 + 22 π}{12}

Add similar elements:

- 3 π + 22 π = 19 π

= 2 πn + \frac{19 π}{12}

3 x = 2 πn + \frac{19 π}{12}

3 x = 2 πn + \frac{19 π}{12}

3 x = 2 πn + \frac{19 π}{12}

Divide both sides by

3

3 x = 2 πn + \frac{19 π}{12}

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{19 π}{12}}{3}

Simplify

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{19 π}{12}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{2 πn}{3} + \frac{\frac{19 π}{12}}{3} : \frac{2 πn}{3} + \frac{19 π}{36}

\frac{2 πn}{3} + \frac{\frac{19 π}{12}}{3}

\frac{\frac{19 π}{12}}{3} = \frac{19 π}{36}

\frac{\frac{19 π}{12}}{3}

Apply the fraction rule:

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

= \frac{19 π}{12 \cdot 3}

Multiply the numbers:

12 \cdot 3 = 36

= \frac{19 π}{36}

= \frac{2 πn}{3} + \frac{19 π}{36}

x = \frac{2 πn}{3} + \frac{19 π}{36}

x = \frac{2 πn}{3} + \frac{19 π}{36}

x = \frac{2 πn}{3} + \frac{19 π}{36}

x = \frac{2 πn}{3} + \frac{11 π}{36}, x = \frac{2 πn}{3} + \frac{19 π}{36}

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