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Popular Trigonometry >

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

Solution

solve for

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

Solution

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

Solution steps

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

Subtract

sin (x) cos (y)

from both sides

sin (x) + cos (y) - cos (y) sin (x) = 0

Rewrite using trig identities

cos (y) + sin (x) - cos (y) sin (x)

Use the basic trigonometric identity:

sin (x) = \frac{1}{csc ( x )}

= cos (y) + \frac{1}{csc ( x )} - cos (y) \frac{1}{csc ( x )}

Simplify

cos (y) + \frac{1}{csc ( x )} - cos (y) \frac{1}{csc ( x )} : cos (y) + \frac{1 - cos ( y )}{csc ( x )}

cos (y) + \frac{1}{csc ( x )} - cos (y) \frac{1}{csc ( x )}

cos (y) \frac{1}{csc ( x )} = \frac{cos ( y )}{csc ( x )}

cos (y) \frac{1}{csc ( x )}

Multiply fractions:

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

= \frac{1 \cdot cos ( y )}{csc ( x )}

Multiply:

1 \cdot cos (y) = cos (y)

= \frac{cos ( y )}{csc ( x )}

= cos (y) + \frac{1}{csc ( x )} - \frac{cos ( y )}{csc ( x )}

Combine the fractions

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

Apply rule

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

= \frac{1 - cos ( y )}{csc ( x )}

= cos (y) + \frac{- cos ( y ) + 1}{csc ( x )}

= cos (y) + \frac{1 - cos ( y )}{csc ( x )}

cos (y) + \frac{1 - cos ( y )}{csc ( x )} = 0

Multiply both sides by

csc (x)

cos (y) + \frac{1 - cos ( y )}{csc ( x )} = 0

Multiply both sides by

csc (x)

cos (y) csc (x) + \frac{1 - cos ( y )}{csc ( x )} csc (x) = 0 \cdot csc (x)

Simplify

cos (y) csc (x) + \frac{1 - cos ( y )}{csc ( x )} csc (x) = 0 \cdot csc (x)

Simplify

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

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

Multiply fractions:

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

= \frac{( 1 - cos ( y ) ) csc ( x )}{csc ( x )}

Cancel the common factor:

csc (x)

= 1 - cos (y)

Simplify

0 \cdot csc (x) : 0

0 \cdot csc (x)

Apply rule

0 \cdot a = 0

= 0

cos (y) csc (x) + 1 - cos (y) = 0

cos (y) csc (x) + 1 - cos (y) = 0

cos (y) csc (x) + 1 - cos (y) = 0

Move

cos (y)

to the right side

cos (y) csc (x) + 1 - cos (y) = 0

Add

cos (y)

to both sides

cos (y) csc (x) + 1 - cos (y) + cos (y) = 0 + cos (y)

Simplify

cos (y) csc (x) + 1 = cos (y)

cos (y) csc (x) + 1 = cos (y)

Move

1

to the right side

cos (y) csc (x) + 1 = cos (y)

Subtract

1

from both sides

cos (y) csc (x) + 1 - 1 = cos (y) - 1

Simplify

cos (y) csc (x) = cos (y) - 1

cos (y) csc (x) = cos (y) - 1

Divide both sides by

cos (y); y \neq = \frac{π}{2} + 2 πn, y \neq = \frac{3 π}{2} + 2 πn

cos (y) csc (x) = cos (y) - 1

Divide both sides by

cos (y); y \neq = \frac{π}{2} + 2 πn, y \neq = \frac{3 π}{2} + 2 πn

\frac{cos ( y ) csc ( x )}{cos ( y )} = \frac{cos ( y )}{cos ( y )} - \frac{1}{cos ( y )}; y \neq = \frac{π}{2} + 2 πn, y \neq = \frac{3 π}{2} + 2 πn

Simplify

\frac{cos ( y ) csc ( x )}{cos ( y )} = \frac{cos ( y )}{cos ( y )} - \frac{1}{cos ( y )}

Simplify

\frac{cos ( y ) csc ( x )}{cos ( y )} : csc (x)

\frac{cos ( y ) csc ( x )}{cos ( y )}

Cancel the common factor:

cos (y)

= csc (x)

Simplify

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

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

Apply rule

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

= \frac{cos ( y ) - 1}{cos ( y )}

csc (x) = \frac{cos ( y ) - 1}{cos ( y )}; y \neq = \frac{π}{2} + 2 πn, y \neq = \frac{3 π}{2} + 2 πn

csc (x) = \frac{cos ( y ) - 1}{cos ( y )}; y \neq = \frac{π}{2} + 2 πn, y \neq = \frac{3 π}{2} + 2 πn

csc (x) = \frac{cos ( y ) - 1}{cos ( y )}; y \neq = \frac{π}{2} + 2 πn, y \neq = \frac{3 π}{2} + 2 πn

Apply trig inverse properties

csc (x) = \frac{cos ( y ) - 1}{cos ( y )}

General solutions for

csc (x) = \frac{cos ( y ) - 1}{cos ( y )}

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π + arccsc (a) + 2 πn

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

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

Graph

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