What is the general solution for sin(x+pi/2)+sin(x-(3pi)/2)=1 ?

The general solution for sin(x+pi/2)+sin(x-(3pi)/2)=1 is x=2pin+pi/2+(7pi)/6 ,x=2pin+pi/2+(11pi)/6

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sin(x+pi/2)+sin(x-(3pi)/2)=1

Solution

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

Solution

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

Degrees

x = 30 0^{\circ} + 36 0^{\circ} n, x = 42 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Use the Sum to Product identity:

sin (s) + sin (t) = 2 sin (\frac{s + t}{2}) cos (\frac{s - t}{2})

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

Simplify

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

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

\frac{x + \frac{π}{2} + x - \frac{3 π}{2}}{2} = \frac{2 x - π}{2}

\frac{x + \frac{π}{2} + x - \frac{3 π}{2}}{2}

Combine the fractions

\frac{π}{2} - \frac{3 π}{2} : - π

Apply rule

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

= \frac{π - 3 π}{2}

Add similar elements:

π - 3 π = - 2 π

= \frac{- 2 π}{2}

Apply the fraction rule:

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

= - \frac{2 π}{2}

Divide the numbers:

\frac{2}{2} = 1

= - π

= \frac{x - π + x}{2}

x - π + x = 2 x - π

x - π + x

Group like terms

= x + x - π

Add similar elements:

x + x = 2 x

= 2 x - π

= \frac{2 x - π}{2}

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

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

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

Join

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

x + \frac{π}{2} - (x - \frac{3 π}{2})

Convert element to fraction:

x = \frac{x 2}{2}, (x - \frac{3 π}{2}) = \frac{( x - \frac{3 π}{2} ) 2}{2}

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

Since the denominators are equal, combine the fractions:

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

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

Expand

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

x \cdot 2 + π - (x - \frac{3 π}{2}) \cdot 2

= 2 x + π - 2 (x - \frac{3 π}{2})

Expand

- 2 (x - \frac{3 π}{2}) : - 2 x + 3 π

- 2 (x - \frac{3 π}{2})

Apply the distributive law:

a (b - c) = ab - a c

a = - 2, b = x, c = \frac{3 π}{2}

= - 2 x - (- 2) \frac{3 π}{2}

Apply minus-plus rules

- (- a) = a

= - 2 x + 2 \cdot \frac{3 π}{2}

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 2}{2}

Cancel the common factor:

2

= 3 π

= - 2 x + 3 π

= x \cdot 2 + π - 2 x + 3 π

Simplify

x \cdot 2 + π - 2 x + 3 π : 4 π

x \cdot 2 + π - 2 x + 3 π

Group like terms

= 2 x - 2 x + π + 3 π

Add similar elements:

2 x - 2 x = 0

= π + 3 π

Add similar elements:

π + 3 π = 4 π

= 4 π

= 4 π

= \frac{4 π}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 π

= \frac{2 π}{2}

Divide the numbers:

\frac{2}{2} = 1

= π

= 2 cos (π) sin (\frac{2 x - π}{2})

Simplify

cos (π) : - 1

cos (π)

Use the following trivial identity:

cos (π) = (- 1)

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - 1

= 2 (- 1) sin (\frac{2 x - π}{2})

Refine

= - 2 sin (\frac{2 x - π}{2})

= - 2 sin (\frac{2 x - π}{2})

- 2 sin (\frac{2 x - π}{2}) = 1

Divide both sides by

- 2

- 2 sin (\frac{2 x - π}{2}) = 1

Divide both sides by

- 2

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

Simplify

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

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

General solutions for

sin (\frac{2 x - π}{2}) = - \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}

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

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

Solve

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

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

Multiply both sides by

2

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

Multiply both sides by

2

\frac{2 ( 2 x - π )}{2} = 2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 ( 2 x - π )}{2} = 2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 ( 2 x - π )}{2} : 2 x - π

\frac{2 ( 2 x - π )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2 x - π

Simplify

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

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{7 π}{6} = \frac{7 π}{3}

2 \cdot \frac{7 π}{6}

Multiply fractions:

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

= \frac{7 π 2}{6}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{14 π}{6}

Cancel the common factor:

2

= \frac{7 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{7 π}{3} + 4 πn

2 x - π = \frac{7 π}{3} + 4 πn

2 x - π = \frac{7 π}{3} + 4 πn

2 x - π = \frac{7 π}{3} + 4 πn

Move

π

to the right side

2 x - π = \frac{7 π}{3} + 4 πn

Add

π

to both sides

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

Simplify

2 x = \frac{7 π}{3} + 4 πn + π

2 x = \frac{7 π}{3} + 4 πn + π

Divide both sides by

2

2 x = \frac{7 π}{3} + 4 πn + π

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

\frac{\frac{7 π}{3}}{2} + \frac{4 πn}{2} + \frac{π}{2}

Group like terms

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

\frac{4 πn}{2} = 2 πn

\frac{4 πn}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 πn

\frac{\frac{7 π}{3}}{2} = \frac{7 π}{6}

\frac{\frac{7 π}{3}}{2}

Apply the fraction rule:

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

= \frac{7 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{7 π}{6}

= \frac{π}{2} + 2 πn + \frac{7 π}{6}

Group like terms

= 2 πn + \frac{π}{2} + \frac{7 π}{6}

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

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

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

Solve

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

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

Multiply both sides by

2

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

Multiply both sides by

2

\frac{2 ( 2 x - π )}{2} = 2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 ( 2 x - π )}{2} = 2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 ( 2 x - π )}{2} : 2 x - π

\frac{2 ( 2 x - π )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2 x - π

Simplify

2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn : \frac{11 π}{3} + 4 πn

2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{11 π}{6} = \frac{11 π}{3}

2 \cdot \frac{11 π}{6}

Multiply fractions:

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

= \frac{11 π 2}{6}

Multiply the numbers:

11 \cdot 2 = 22

= \frac{22 π}{6}

Cancel the common factor:

2

= \frac{11 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{11 π}{3} + 4 πn

2 x - π = \frac{11 π}{3} + 4 πn

2 x - π = \frac{11 π}{3} + 4 πn

2 x - π = \frac{11 π}{3} + 4 πn

Move

π

to the right side

2 x - π = \frac{11 π}{3} + 4 πn

Add

π

to both sides

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

Simplify

2 x = \frac{11 π}{3} + 4 πn + π

2 x = \frac{11 π}{3} + 4 πn + π

Divide both sides by

2

2 x = \frac{11 π}{3} + 4 πn + π

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

\frac{\frac{11 π}{3}}{2} + \frac{4 πn}{2} + \frac{π}{2}

Group like terms

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

\frac{4 πn}{2} = 2 πn

\frac{4 πn}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 πn

\frac{\frac{11 π}{3}}{2} = \frac{11 π}{6}

\frac{\frac{11 π}{3}}{2}

Apply the fraction rule:

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

= \frac{11 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{11 π}{6}

= \frac{π}{2} + 2 πn + \frac{11 π}{6}

Group like terms

= 2 πn + \frac{π}{2} + \frac{11 π}{6}

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

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

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

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

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x+pi/2)+sin(x-(3pi)/2)=1 ?
The general solution for sin(x+pi/2)+sin(x-(3pi)/2)=1 is x=2pin+pi/2+(7pi)/6 ,x=2pin+pi/2+(11pi)/6