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Popular Trigonometry >

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

Solution

solve for

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

Solution

x = \frac{π + 4 πn - 2 y}{2 ( 1 + y )}, x = \frac{π + 4 πn - 2 y}{2 ( 1 - y )}

Solution steps

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

Rewrite using trig identities

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

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

cos (x y) = sin (\frac{π}{2} - x y)

cos (x y) = sin (\frac{π}{2} - x y)

Apply trig inverse properties

cos (x y) = sin (\frac{π}{2} - x y)

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

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

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

x + y = \frac{π}{2} - x y + 2 πn : x = \frac{π + 4 πn - 2 y}{2 ( 1 + y )}; y \neq = - 1

x + y = \frac{π}{2} - x y + 2 πn

Move

y

to the right side

x + y = \frac{π}{2} - x y + 2 πn

Subtract

y

from both sides

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

Simplify

x = \frac{π}{2} - x y + 2 πn - y

x = \frac{π}{2} - x y + 2 πn - y

Move

x y

to the left side

x = \frac{π}{2} - x y + 2 πn - y

Add

x y

to both sides

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

Simplify

x + x y = \frac{π}{2} + 2 πn - y

x + x y = \frac{π}{2} + 2 πn - y

Factor

x + x y : x (1 + y)

x + x y

Factor out common term

x

= x (1 + y)

x (1 + y) = \frac{π}{2} + 2 πn - y

Divide both sides by

1 + y; y \neq = - 1

x (1 + y) = \frac{π}{2} + 2 πn - y

Divide both sides by

1 + y; y \neq = - 1

\frac{x ( 1 + y )}{1 + y} = \frac{\frac{π}{2}}{1 + y} + \frac{2 πn}{1 + y} - \frac{y}{1 + y}; y \neq = - 1

Simplify

\frac{x ( 1 + y )}{1 + y} = \frac{\frac{π}{2}}{1 + y} + \frac{2 πn}{1 + y} - \frac{y}{1 + y}

Simplify

\frac{x ( 1 + y )}{1 + y} : x

\frac{x ( 1 + y )}{1 + y}

Cancel the common factor:

1 + y

= x

Simplify

\frac{\frac{π}{2}}{1 + y} + \frac{2 πn}{1 + y} - \frac{y}{1 + y} : \frac{π + 4 πn - 2 y}{2 ( 1 + y )}

\frac{\frac{π}{2}}{1 + y} + \frac{2 πn}{1 + y} - \frac{y}{1 + y}

Apply rule

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

= \frac{\frac{π}{2} + 2 πn - y}{1 + y}

Join

\frac{π}{2} + 2 πn - y : \frac{π + 4 πn - 2 y}{2}

\frac{π}{2} + 2 πn - y

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, y = \frac{y 2}{2}

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π + 4 πn - 2 y}{2}

= \frac{\frac{π + 4 πn - 2 y}{2}}{1 + y}

Apply the fraction rule:

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

= \frac{π + 4 πn - y \cdot 2}{2 ( 1 + y )}

x = \frac{π + 4 πn - 2 y}{2 ( 1 + y )}; y \neq = - 1

x = \frac{π + 4 πn - 2 y}{2 ( 1 + y )}; y \neq = - 1

x = \frac{π + 4 πn - 2 y}{2 ( 1 + y )}; y \neq = - 1

x + y = π - (\frac{π}{2} - x y) + 2 πn : x = \frac{π + 4 πn - 2 y}{2 ( 1 - y )}; y \neq = 1

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

Expand

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

π - (\frac{π}{2} - x y) + 2 πn

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

- (\frac{π}{2} - x y)

Distribute parentheses

= - \frac{π}{2} - (- x y)

Apply minus-plus rules

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

= - \frac{π}{2} + x y

= π - \frac{π}{2} + x y + 2 πn

x + y = π - \frac{π}{2} + x y + 2 πn

Move

y

to the right side

x + y = π - \frac{π}{2} + x y + 2 πn

Subtract

y

from both sides

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

Simplify

x = π - \frac{π}{2} + x y + 2 πn - y

x = π - \frac{π}{2} + x y + 2 πn - y

Move

x y

to the left side

x = π - \frac{π}{2} + x y + 2 πn - y

Subtract

x y

from both sides

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

Simplify

x - x y = π - \frac{π}{2} + 2 πn - y

x - x y = π - \frac{π}{2} + 2 πn - y

Factor

x - x y : x (1 - y)

x - x y

Factor out common term

x

= x (1 - y)

x (1 - y) = π - \frac{π}{2} + 2 πn - y

Divide both sides by

1 - y; y \neq = 1

x (1 - y) = π - \frac{π}{2} + 2 πn - y

Divide both sides by

1 - y; y \neq = 1

\frac{x ( 1 - y )}{1 - y} = \frac{π}{1 - y} - \frac{\frac{π}{2}}{1 - y} + \frac{2 πn}{1 - y} - \frac{y}{1 - y}; y \neq = 1

Simplify

\frac{x ( 1 - y )}{1 - y} = \frac{π}{1 - y} - \frac{\frac{π}{2}}{1 - y} + \frac{2 πn}{1 - y} - \frac{y}{1 - y}

Simplify

\frac{x ( 1 - y )}{1 - y} : x

\frac{x ( 1 - y )}{1 - y}

Cancel the common factor:

1 - y

= x

Simplify

\frac{π}{1 - y} - \frac{\frac{π}{2}}{1 - y} + \frac{2 πn}{1 - y} - \frac{y}{1 - y} : \frac{π + 4 πn - 2 y}{2 ( 1 - y )}

\frac{π}{1 - y} - \frac{\frac{π}{2}}{1 - y} + \frac{2 πn}{1 - y} - \frac{y}{1 - y}

Apply rule

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

= \frac{π - \frac{π}{2} + 2 πn - y}{1 - y}

Join

π - \frac{π}{2} + 2 πn - y : \frac{π + 4 πn - 2 y}{2}

π - \frac{π}{2} + 2 πn - y

Convert element to fraction:

π = \frac{π 2}{2}, 2 πn = \frac{2 πn 2}{2}, y = \frac{y 2}{2}

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

Since the denominators are equal, combine the fractions:

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

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

π 2 - π + 2 πn \cdot 2 - y \cdot 2 = π + 4 πn - 2 y

π 2 - π + 2 πn \cdot 2 - y \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn - 2 y

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn - 2 y

= \frac{π + 4 πn - 2 y}{2}

= \frac{\frac{π + 4 πn - 2 y}{2}}{1 - y}

Apply the fraction rule:

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

= \frac{π + 4 πn - y \cdot 2}{2 ( 1 - y )}

x = \frac{π + 4 πn - 2 y}{2 ( 1 - y )}; y \neq = 1

x = \frac{π + 4 πn - 2 y}{2 ( 1 - y )}; y \neq = 1

x = \frac{π + 4 πn - 2 y}{2 ( 1 - y )}; y \neq = 1

x = \frac{π + 4 πn - 2 y}{2 ( 1 + y )}, x = \frac{π + 4 πn - 2 y}{2 ( 1 - y )}

x = \frac{π + 4 πn - 2 y}{2 ( 1 + y )}, x = \frac{π + 4 πn - 2 y}{2 ( 1 - y )}

Graph

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