What is the general solution for 4sin(4θ-pi/3)+3=5 ?

The general solution for 4sin(4θ-pi/3)+3=5 is θ=(pin)/2+pi/8 ,θ=(pin)/2+(7pi)/(24)

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Popular Trigonometry >

4sin(4θ-pi/3)+3=5

Solution

4 sin (4 θ - \frac{π}{3}) + 3 = 5

Solution

θ = \frac{πn}{2} + \frac{π}{8}, θ = \frac{πn}{2} + \frac{7 π}{24}

Degrees

θ = 22. 5^{\circ} + 9 0^{\circ} n, θ = 52. 5^{\circ} + 9 0^{\circ} n

Solution steps

4 sin (4 θ - \frac{π}{3}) + 3 = 5

Move

3

to the right side

4 sin (4 θ - \frac{π}{3}) + 3 = 5

Subtract

3

from both sides

4 sin (4 θ - \frac{π}{3}) + 3 - 3 = 5 - 3

Simplify

4 sin (4 θ - \frac{π}{3}) = 2

4 sin (4 θ - \frac{π}{3}) = 2

Divide both sides by

4

4 sin (4 θ - \frac{π}{3}) = 2

Divide both sides by

4

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

Simplify

sin (4 θ - \frac{π}{3}) = \frac{1}{2}

sin (4 θ - \frac{π}{3}) = \frac{1}{2}

General solutions for

sin (4 θ - \frac{π}{3}) = \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}

4 θ - \frac{π}{3} = \frac{π}{6} + 2 πn, 4 θ - \frac{π}{3} = \frac{5 π}{6} + 2 πn

4 θ - \frac{π}{3} = \frac{π}{6} + 2 πn, 4 θ - \frac{π}{3} = \frac{5 π}{6} + 2 πn

Solve

4 θ - \frac{π}{3} = \frac{π}{6} + 2 πn : θ = \frac{πn}{2} + \frac{π}{8}

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

Move

\frac{π}{3}

to the right side

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

Add

\frac{π}{3}

to both sides

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

Simplify

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

Simplify

4 θ - \frac{π}{3} + \frac{π}{3} : 4 θ

4 θ - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} = 0

= 4 θ

Simplify

\frac{π}{6} + 2 πn + \frac{π}{3} : 2 πn + \frac{π}{2}

\frac{π}{6} + 2 πn + \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{6} + \frac{π}{3}

Least Common Multiplier of

6, 3 : 6

6, 3

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

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

6

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

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

6

For

\frac{π}{3} :

multiply the denominator and numerator by

2

\frac{π}{3} = \frac{π 2}{3 \cdot 2} = \frac{π 2}{6}

= \frac{π}{6} + \frac{π 2}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π + π 2}{6}

Add similar elements:

π + 2 π = 3 π

= \frac{3 π}{6}

Cancel the common factor:

3

= 2 πn + \frac{π}{2}

4 θ = 2 πn + \frac{π}{2}

4 θ = 2 πn + \frac{π}{2}

4 θ = 2 πn + \frac{π}{2}

Divide both sides by

4

4 θ = 2 πn + \frac{π}{2}

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 θ}{4} : θ

\frac{4 θ}{4}

Divide the numbers:

\frac{4}{4} = 1

= θ

Simplify

\frac{2 πn}{4} + \frac{\frac{π}{2}}{4} : \frac{πn}{2} + \frac{π}{8}

\frac{2 πn}{4} + \frac{\frac{π}{2}}{4}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

\frac{\frac{π}{2}}{4} = \frac{π}{8}

\frac{\frac{π}{2}}{4}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{π}{8}

= \frac{πn}{2} + \frac{π}{8}

θ = \frac{πn}{2} + \frac{π}{8}

θ = \frac{πn}{2} + \frac{π}{8}

θ = \frac{πn}{2} + \frac{π}{8}

Solve

4 θ - \frac{π}{3} = \frac{5 π}{6} + 2 πn : θ = \frac{πn}{2} + \frac{7 π}{24}

4 θ - \frac{π}{3} = \frac{5 π}{6} + 2 πn

Move

\frac{π}{3}

to the right side

4 θ - \frac{π}{3} = \frac{5 π}{6} + 2 πn

Add

\frac{π}{3}

to both sides

4 θ - \frac{π}{3} + \frac{π}{3} = \frac{5 π}{6} + 2 πn + \frac{π}{3}

Simplify

4 θ - \frac{π}{3} + \frac{π}{3} = \frac{5 π}{6} + 2 πn + \frac{π}{3}

Simplify

4 θ - \frac{π}{3} + \frac{π}{3} : 4 θ

4 θ - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} = 0

= 4 θ

Simplify

\frac{5 π}{6} + 2 πn + \frac{π}{3} : 2 πn + \frac{7 π}{6}

\frac{5 π}{6} + 2 πn + \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{3} + \frac{5 π}{6}

Least Common Multiplier of

3, 6 : 6

3, 6

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

3

6

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

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

6

For

\frac{π}{3} :

multiply the denominator and numerator by

2

\frac{π}{3} = \frac{π 2}{3 \cdot 2} = \frac{π 2}{6}

= \frac{π 2}{6} + \frac{5 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 + 5 π}{6}

Add similar elements:

2 π + 5 π = 7 π

= 2 πn + \frac{7 π}{6}

4 θ = 2 πn + \frac{7 π}{6}

4 θ = 2 πn + \frac{7 π}{6}

4 θ = 2 πn + \frac{7 π}{6}

Divide both sides by

4

4 θ = 2 πn + \frac{7 π}{6}

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 θ}{4} : θ

\frac{4 θ}{4}

Divide the numbers:

\frac{4}{4} = 1

= θ

Simplify

\frac{2 πn}{4} + \frac{\frac{7 π}{6}}{4} : \frac{πn}{2} + \frac{7 π}{24}

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

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

\frac{\frac{7 π}{6}}{4} = \frac{7 π}{24}

\frac{\frac{7 π}{6}}{4}

Apply the fraction rule:

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

= \frac{7 π}{6 \cdot 4}

Multiply the numbers:

6 \cdot 4 = 24

= \frac{7 π}{24}

= \frac{πn}{2} + \frac{7 π}{24}

θ = \frac{πn}{2} + \frac{7 π}{24}

θ = \frac{πn}{2} + \frac{7 π}{24}

θ = \frac{πn}{2} + \frac{7 π}{24}

θ = \frac{πn}{2} + \frac{π}{8}, θ = \frac{πn}{2} + \frac{7 π}{24}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 4sin(4θ-pi/3)+3=5 ?
The general solution for 4sin(4θ-pi/3)+3=5 is θ=(pin)/2+pi/8 ,θ=(pin)/2+(7pi)/(24)