What is the general solution for -6=-2+8sin(θ+pi/4) ?

The general solution for -6=-2+8sin(θ+pi/4) is θ=2pin+(11pi)/(12),θ=2pin+(19pi)/(12)

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Popular >

-6=-2+8sin(θ+pi/4)

Solution

- 6 = - 2 + 8 sin (θ + \frac{π}{4})

Solution

θ = 2 πn + \frac{11 π}{12}, θ = 2 πn + \frac{19 π}{12}

Degrees

θ = 16 5^{\circ} + 36 0^{\circ} n, θ = 28 5^{\circ} + 36 0^{\circ} n

Solution steps

- 6 = - 2 + 8 sin (θ + \frac{π}{4})

Switch sides

- 2 + 8 sin (θ + \frac{π}{4}) = - 6

Move

2

to the right side

- 2 + 8 sin (θ + \frac{π}{4}) = - 6

Add

2

to both sides

- 2 + 8 sin (θ + \frac{π}{4}) + 2 = - 6 + 2

Simplify

8 sin (θ + \frac{π}{4}) = - 4

8 sin (θ + \frac{π}{4}) = - 4

Divide both sides by

8

8 sin (θ + \frac{π}{4}) = - 4

Divide both sides by

8

\frac{8 sin ( θ + \frac{π}{4} )}{8} = \frac{- 4}{8}

Simplify

\frac{8 sin ( θ + \frac{π}{4} )}{8} = \frac{- 4}{8}

Simplify

\frac{8 sin ( θ + \frac{π}{4} )}{8} : sin (θ + \frac{π}{4})

\frac{8 sin ( θ + \frac{π}{4} )}{8}

Divide the numbers:

\frac{8}{8} = 1

= sin (θ + \frac{π}{4})

Simplify

\frac{- 4}{8} : - \frac{1}{2}

\frac{- 4}{8}

Apply the fraction rule:

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

= - \frac{4}{8}

Cancel the common factor:

4

= - \frac{1}{2}

sin (θ + \frac{π}{4}) = - \frac{1}{2}

sin (θ + \frac{π}{4}) = - \frac{1}{2}

sin (θ + \frac{π}{4}) = - \frac{1}{2}

General solutions for

sin (θ + \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}

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

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

Solve

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

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

Simplify

θ + \frac{π}{4} - \frac{π}{4} : θ

θ + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= θ

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}

θ = 2 πn + \frac{11 π}{12}

θ = 2 πn + \frac{11 π}{12}

θ = 2 πn + \frac{11 π}{12}

Solve

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

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

Simplify

θ + \frac{π}{4} - \frac{π}{4} : θ

θ + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= θ

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}

θ = 2 πn + \frac{19 π}{12}

θ = 2 πn + \frac{19 π}{12}

θ = 2 πn + \frac{19 π}{12}

θ = 2 πn + \frac{11 π}{12}, θ = 2 πn + \frac{19 π}{12}

Popular Examples

0=12cos(pi/3 x)+8

0 = 12 cos (\frac{π}{3} x) + 8

0=sin(t)+2sin(2t)

0 = sin (t) + 2 sin (2 t)

sin(x)=-0.9

sin (x) = - 0.9

1=tan(pi/4+c)

1 = tan (\frac{π}{4} + c)

cos(θ)-cos(3θ)=0

cos (θ) - cos (3 θ) = 0

Frequently Asked Questions (FAQ)

What is the general solution for -6=-2+8sin(θ+pi/4) ?
The general solution for -6=-2+8sin(θ+pi/4) is θ=2pin+(11pi)/(12),θ=2pin+(19pi)/(12)