What is the general solution for 2sin(x-pi/2)+sqrt(2)=0 ?

The general solution for 2sin(x-pi/2)+sqrt(2)=0 is x=2pin+(7pi)/4 ,x=2pin+(9pi)/4

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2sin(x-pi/2)+sqrt(2)=0

Solution

2 sin (x - \frac{π}{2}) + 2 = 0

Solution

x = 2 πn + \frac{7 π}{4}, x = 2 πn + \frac{9 π}{4}

Degrees

x = 31 5^{\circ} + 36 0^{\circ} n, x = 40 5^{\circ} + 36 0^{\circ} n

Solution steps

2 sin (x - \frac{π}{2}) + 2 = 0

Move

2

to the right side

2 sin (x - \frac{π}{2}) + 2 = 0

Subtract

2

from both sides

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

Simplify

2 sin (x - \frac{π}{2}) = - 2

2 sin (x - \frac{π}{2}) = - 2

Divide both sides by

2

2 sin (x - \frac{π}{2}) = - 2

Divide both sides by

2

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

Simplify

sin (x - \frac{π}{2}) = - \frac{2}{2}

sin (x - \frac{π}{2}) = - \frac{2}{2}

General solutions for

sin (x - \frac{π}{2}) = - \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}

x - \frac{π}{2} = \frac{5 π}{4} + 2 πn, x - \frac{π}{2} = \frac{7 π}{4} + 2 πn

x - \frac{π}{2} = \frac{5 π}{4} + 2 πn, x - \frac{π}{2} = \frac{7 π}{4} + 2 πn

Solve

x - \frac{π}{2} = \frac{5 π}{4} + 2 πn : x = 2 πn + \frac{7 π}{4}

x - \frac{π}{2} = \frac{5 π}{4} + 2 πn

Move

\frac{π}{2}

to the right side

x - \frac{π}{2} = \frac{5 π}{4} + 2 πn

Add

\frac{π}{2}

to both sides

x - \frac{π}{2} + \frac{π}{2} = \frac{5 π}{4} + 2 πn + \frac{π}{2}

Simplify

x - \frac{π}{2} + \frac{π}{2} = \frac{5 π}{4} + 2 πn + \frac{π}{2}

Simplify

x - \frac{π}{2} + \frac{π}{2} : x

x - \frac{π}{2} + \frac{π}{2}

Add similar elements:

- \frac{π}{2} + \frac{π}{2} = 0

= x

Simplify

\frac{5 π}{4} + 2 πn + \frac{π}{2} : 2 πn + \frac{7 π}{4}

\frac{5 π}{4} + 2 πn + \frac{π}{2}

Group like terms

= 2 πn + \frac{π}{2} + \frac{5 π}{4}

Least Common Multiplier of

2, 4 : 4

2, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

2

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{π}{2} :

multiply the denominator and numerator by

2

\frac{π}{2} = \frac{π 2}{2 \cdot 2} = \frac{π 2}{4}

= \frac{π 2}{4} + \frac{5 π}{4}

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}{4}

Add similar elements:

2 π + 5 π = 7 π

= 2 πn + \frac{7 π}{4}

x = 2 πn + \frac{7 π}{4}

x = 2 πn + \frac{7 π}{4}

x = 2 πn + \frac{7 π}{4}

Solve

x - \frac{π}{2} = \frac{7 π}{4} + 2 πn : x = 2 πn + \frac{9 π}{4}

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

Move

\frac{π}{2}

to the right side

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

Add

\frac{π}{2}

to both sides

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

Simplify

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

Simplify

x - \frac{π}{2} + \frac{π}{2} : x

x - \frac{π}{2} + \frac{π}{2}

Add similar elements:

- \frac{π}{2} + \frac{π}{2} = 0

= x

Simplify

\frac{7 π}{4} + 2 πn + \frac{π}{2} : 2 πn + \frac{9 π}{4}

\frac{7 π}{4} + 2 πn + \frac{π}{2}

Group like terms

= 2 πn + \frac{π}{2} + \frac{7 π}{4}

Least Common Multiplier of

2, 4 : 4

2, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

2

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{π}{2} :

multiply the denominator and numerator by

2

\frac{π}{2} = \frac{π 2}{2 \cdot 2} = \frac{π 2}{4}

= \frac{π 2}{4} + \frac{7 π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 + 7 π}{4}

Add similar elements:

2 π + 7 π = 9 π

= 2 πn + \frac{9 π}{4}

x = 2 πn + \frac{9 π}{4}

x = 2 πn + \frac{9 π}{4}

x = 2 πn + \frac{9 π}{4}

x = 2 πn + \frac{7 π}{4}, x = 2 πn + \frac{9 π}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin(x-pi/2)+sqrt(2)=0 ?
The general solution for 2sin(x-pi/2)+sqrt(2)=0 is x=2pin+(7pi)/4 ,x=2pin+(9pi)/4