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

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

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cos((x+pi)/2)=sin(x)

Solution

cos (\frac{x + π}{2}) = sin (x)

Solution

x = \frac{4 πn}{3}, x = 2 π + 4 πn

Degrees

x = 0^{\circ} + 24 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n

Solution steps

cos (\frac{x + π}{2}) = sin (x)

Rewrite using trig identities

cos (\frac{x + π}{2}) = sin (x)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

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

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

Apply trig inverse properties

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

sin (x) = sin (y) \Rightarrow x = y + 2 π n, x = π - y + 2 π n

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

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

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

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

Simplify

x \cdot 2 : 2 x

x \cdot 2

Apply the commutative law:

x \cdot 2 = 2 x

2 x

Simplify

\frac{π}{2} \cdot 2 : π

\frac{π}{2} \cdot 2

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

Simplify

- \frac{x + π}{2} \cdot 2 : - (x + π)

- \frac{x + π}{2} \cdot 2

Multiply fractions:

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

= - \frac{( x + π ) \cdot 2}{2}

Cancel the common factor:

2

= - (x + π)

Simplify

2 πn \cdot 2 : 4 πn

2 πn \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 x = π - (x + π) + 4 πn

2 x = π - (x + π) + 4 πn

2 x = π - (x + π) + 4 πn

Expand

π - (x + π) + 4 πn : - x + 4 πn

π - (x + π) + 4 πn

- (x + π) : - x - π

- (x + π)

Distribute parentheses

= - (x) - (π)

Apply minus-plus rules

+ (- a) = - a

= - x - π

= π - x - π + 4 πn

Simplify

π - x - π + 4 πn : - x + 4 πn

π - x - π + 4 πn

Group like terms

= - x + π - π + 4 πn

Add similar elements:

π - π = 0

= - x + 4 πn

= - x + 4 πn

2 x = - x + 4 πn

Move

x

to the left side

2 x = - x + 4 πn

Add

x

to both sides

2 x + x = - x + 4 πn + x

Simplify

3 x = 4 πn

3 x = 4 πn

Divide both sides by

3

3 x = 4 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{4 πn}{3}

Simplify

x = \frac{4 πn}{3}

x = \frac{4 πn}{3}

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

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

Move

(\frac{π}{2} - \frac{x + π}{2})

to the left side

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

Add

(\frac{π}{2} - \frac{x + π}{2})

to both sides

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

Simplify

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

Simplify

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

x + \frac{π}{2} - \frac{x + π}{2}

Combine the fractions

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

Apply rule

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

= \frac{π - ( x + π )}{2}

= x + \frac{π - ( x + π )}{2}

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

\frac{π - ( x + π )}{2}

Expand

π - (x + π) : - x

π - (x + π)

- (x + π) : - x - π

- (x + π)

Distribute parentheses

= - (x) - (π)

Apply minus-plus rules

+ (- a) = - a

= - x - π

= π - x - π

Simplify

π - x - π : - x

π - x - π

Group like terms

= - x + π - π

Add similar elements:

π - π = 0

= - x

= - x

= \frac{- x}{2}

Apply the fraction rule:

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

= - \frac{x}{2}

= x - \frac{x}{2}

Convert element to fraction:

x = \frac{x 2}{2}

= - \frac{x}{2} + \frac{x \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 + x \cdot 2}{2}

Add similar elements:

- x + 2 x = x

= \frac{x}{2}

Simplify

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

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

Add similar elements:

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

= π + 2 πn

\frac{x}{2} = π + 2 πn

\frac{x}{2} = π + 2 πn

\frac{x}{2} = π + 2 πn

Multiply both sides by

2

\frac{x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 π + 2 \cdot 2 πn

Simplify

x = 2 π + 4 πn

x = 2 π + 4 πn

x = \frac{4 πn}{3}, x = 2 π + 4 πn

x = \frac{4 πn}{3}, x = 2 π + 4 πn

Graph

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Frequently Asked Questions (FAQ)

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