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

The general solution for sin(x)+sin(pi/2-x)=0 is x=(3pi)/4+2pin,x=(7pi)/4+2pin

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

Solution

sin (x) + sin (\frac{π}{2} - x) = 0

Solution

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

Degrees

x = 13 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

sin (x) + sin (\frac{π}{2} - x) = 0

Rewrite using trig identities

sin (x) + sin (\frac{π}{2} - x)

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}{2}) cos (\frac{x - ( \frac{π}{2} - x )}{2})

Simplify

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

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

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

\frac{x + \frac{π}{2} - x}{2}

x + \frac{π}{2} - x = \frac{π}{2}

x + \frac{π}{2} - x

Group like terms

= x - x + \frac{π}{2}

Add similar elements:

x - x = 0

= \frac{π}{2}

= \frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

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

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

\frac{x - ( \frac{π}{2} - x )}{2}

Expand

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

x - (\frac{π}{2} - x)

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

- (\frac{π}{2} - x)

Distribute parentheses

= - (\frac{π}{2}) - (- x)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - \frac{π}{2} + x

= x - \frac{π}{2} + x

Simplify

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

x - \frac{π}{2} + x

Group like terms

= x + x - \frac{π}{2}

Add similar elements:

x + x = 2 x

= 2 x - \frac{π}{2}

= 2 x - \frac{π}{2}

= \frac{2 x - \frac{π}{2}}{2}

Join

2 x - \frac{π}{2} : \frac{4 x - π}{2}

2 x - \frac{π}{2}

Convert element to fraction:

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

= \frac{2 x \cdot 2}{2} - \frac{π}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{2 x \cdot 2 - π}{2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 x - π}{2}

= \frac{\frac{4 x - π}{2}}{2}

Apply the fraction rule:

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

= \frac{4 x - π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 x - π}{4}

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

Simplify

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Use the following trivial identity:

sin (\frac{π}{4}) = \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}

= \frac{2}{2}

= 2 \cdot \frac{2}{2} cos (\frac{4 x - π}{4})

Multiply fractions:

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

= \frac{2 \cdot 2 cos ( \frac{4 x - π}{4} )}{2}

Cancel the common factor:

2

= 2 cos (\frac{4 x - π}{4})

= 2 cos (\frac{4 x - π}{4})

2 cos (\frac{4 x - π}{4}) = 0

Divide both sides by

2

2 cos (\frac{4 x - π}{4}) = 0

Divide both sides by

2

\frac{2 cos ( \frac{4 x - π}{4} )}{2} = \frac{0}{2}

Simplify

cos (\frac{4 x - π}{4}) = 0

cos (\frac{4 x - π}{4}) = 0

General solutions for

cos (\frac{4 x - π}{4}) = 0

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}

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

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

Solve

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

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

Multiply both sides by

4

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

Multiply both sides by

4

\frac{4 ( 4 x - π )}{4} = 4 \cdot \frac{π}{2} + 4 \cdot 2 πn

Simplify

\frac{4 ( 4 x - π )}{4} = 4 \cdot \frac{π}{2} + 4 \cdot 2 πn

Simplify

\frac{4 ( 4 x - π )}{4} : 4 x - π

\frac{4 ( 4 x - π )}{4}

Divide the numbers:

\frac{4}{4} = 1

= 4 x - π

Simplify

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

4 \cdot \frac{π}{2} + 4 \cdot 2 πn

4 \cdot \frac{π}{2} = 2 π

4 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= 2 π + 8 πn

4 x - π = 2 π + 8 πn

4 x - π = 2 π + 8 πn

4 x - π = 2 π + 8 πn

Move

π

to the right side

4 x - π = 2 π + 8 πn

Add

π

to both sides

4 x - π + π = 2 π + 8 πn + π

Simplify

4 x = 3 π + 8 πn

4 x = 3 π + 8 πn

Divide both sides by

4

4 x = 3 π + 8 πn

Divide both sides by

4

\frac{4 x}{4} = \frac{3 π}{4} + \frac{8 πn}{4}

Simplify

x = \frac{3 π}{4} + 2 πn

x = \frac{3 π}{4} + 2 πn

Solve

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

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

Multiply both sides by

4

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

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 ( 4 x - π )}{4} : 4 x - π

\frac{4 ( 4 x - π )}{4}

Divide the numbers:

\frac{4}{4} = 1

= 4 x - π

Simplify

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn : 6 π + 8 πn

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

4 \cdot \frac{3 π}{2} = 6 π

4 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 4}{2}

Multiply the numbers:

3 \cdot 4 = 12

= \frac{12 π}{2}

Divide the numbers:

\frac{12}{2} = 6

= 6 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= 6 π + 8 πn

4 x - π = 6 π + 8 πn

4 x - π = 6 π + 8 πn

4 x - π = 6 π + 8 πn

Move

π

to the right side

4 x - π = 6 π + 8 πn

Add

π

to both sides

4 x - π + π = 6 π + 8 πn + π

Simplify

4 x = 7 π + 8 πn

4 x = 7 π + 8 πn

Divide both sides by

4

4 x = 7 π + 8 πn

Divide both sides by

4

\frac{4 x}{4} = \frac{7 π}{4} + \frac{8 πn}{4}

Simplify

x = \frac{7 π}{4} + 2 πn

x = \frac{7 π}{4} + 2 πn

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

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)+sin(pi/2-x)=0 ?
The general solution for sin(x)+sin(pi/2-x)=0 is x=(3pi)/4+2pin,x=(7pi)/4+2pin