What is the general solution for solvefor x,sin(x+z)=cos(y+z) ?

The general solution for solvefor x,sin(x+z)=cos(y+z) is x=-2z+2pin+pi/2-y,x=pi+2pin-pi/2+y

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Popular Trigonometry >

solvefor x,sin(x+z)=cos(y+z)

Solution

solvefor

Solution

x = - 2 z + 2 πn + \frac{π}{2} - y, x = π + 2 πn - \frac{π}{2} + y

Solution steps

sin (x + z) = cos (y + z)

Rewrite using trig identities

cos (y + z)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (\frac{π}{2} - (y + z))

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

Apply trig inverse properties

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

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

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

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

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

x + z = \frac{π}{2} - (y + z) + 2 πn

Move

z

to the right side

x + z = \frac{π}{2} - (y + z) + 2 πn

Subtract

z

from both sides

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

Simplify

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

Simplify

x + z - z : x

x + z - z

Add similar elements:

z - z = 0

= x

Simplify

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

\frac{π}{2} - (y + z) + 2 πn - z

- (y + z) : - y - z

- (y + z)

Distribute parentheses

= - y - z

Apply minus-plus rules

+ (- a) = - a

= - y - z

= \frac{π}{2} - y - z + 2 πn - z

Simplify

\frac{π}{2} - y - z + 2 πn - z : - 2 z + 2 πn + \frac{π}{2} - y

\frac{π}{2} - y - z + 2 πn - z

Group like terms

= - z - z + 2 πn + \frac{π}{2} - y

Add similar elements:

- z - z = - 2 z

= - 2 z + 2 πn + \frac{π}{2} - y

= - 2 z + 2 πn + \frac{π}{2} - y

x = - 2 z + 2 πn + \frac{π}{2} - y

x = - 2 z + 2 πn + \frac{π}{2} - y

x = - 2 z + 2 πn + \frac{π}{2} - y

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

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

Move

z

to the right side

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

Subtract

z

from both sides

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

Simplify

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

Simplify

x + z - z : x

x + z - z

Add similar elements:

z - z = 0

= x

Simplify

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

π - (\frac{π}{2} - (y + z)) + 2 πn - z

- (y + z) : - y - z

- (y + z)

Distribute parentheses

= - y - z

Apply minus-plus rules

+ (- a) = - a

= - y - z

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

- (\frac{π}{2} - y - z) : - \frac{π}{2} + y + z

- (\frac{π}{2} - y - z)

Distribute parentheses

= - \frac{π}{2} - (- y) - (- z)

Apply minus-plus rules

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

= - \frac{π}{2} + y + z

= π - \frac{π}{2} + y + z + 2 πn - z

Simplify

π - \frac{π}{2} + y + z + 2 πn - z : π + 2 πn - \frac{π}{2} + y

π - \frac{π}{2} + y + z + 2 πn - z

Group like terms

= z - z + π + 2 πn - \frac{π}{2} + y

Add similar elements:

z - z = 0

= π + 2 πn - \frac{π}{2} + y

= π + 2 πn - \frac{π}{2} + y

x = π + 2 πn - \frac{π}{2} + y

x = π + 2 πn - \frac{π}{2} + y

x = π + 2 πn - \frac{π}{2} + y

x = - 2 z + 2 πn + \frac{π}{2} - y, x = π + 2 πn - \frac{π}{2} + y

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor x,sin(x+z)=cos(y+z) ?
The general solution for solvefor x,sin(x+z)=cos(y+z) is x=-2z+2pin+pi/2-y,x=pi+2pin-pi/2+y