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Popular >

solvefor x,e^{cos(y)}-e^{sin(x)}=sin(y)

Solution

solve for

x, e^{c o s (y)} - e^{s i n (x)} = sin (y)

Solution

x = arcsin (ln (- sin (y) + e^{c o s (y)})) + 2 πn, x = π + arcsin (- ln (- sin (y) + e^{c o s (y)})) + 2 πn

Solution steps

e^{c o s (y)} - e^{s i n (x)} = sin (y)

Move

e^{c o s (y)}

to the right side

e^{c o s (y)} - e^{s i n (x)} = sin (y)

Subtract

e^{c o s (y)}

from both sides

e^{c o s (y)} - e^{s i n (x)} - e^{c o s (y)} = sin (y) - e^{c o s (y)}

Simplify

- e^{s i n (x)} = sin (y) - e^{c o s (y)}

- e^{s i n (x)} = sin (y) - e^{c o s (y)}

Divide both sides by

- 1

- e^{s i n (x)} = sin (y) - e^{c o s (y)}

Divide both sides by

- 1

\frac{- e ^{s i n (x)}}{- 1} = \frac{sin ( y )}{- 1} - \frac{e ^{c o s (y)}}{- 1}

Simplify

\frac{- e ^{s i n (x)}}{- 1} = \frac{sin ( y )}{- 1} - \frac{e ^{c o s (y)}}{- 1}

Simplify

\frac{- e ^{s i n (x)}}{- 1} : e^{s i n (x)}

\frac{- e ^{s i n (x)}}{- 1}

Apply the fraction rule:

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

= \frac{e ^{s i n (x)}}{1}

Apply rule

\frac{a}{1} = a

= e^{s i n (x)}

Simplify

\frac{sin ( y )}{- 1} - \frac{e ^{c o s (y)}}{- 1} : - sin (y) + e^{c o s (y)}

\frac{sin ( y )}{- 1} - \frac{e ^{c o s (y)}}{- 1}

Apply rule

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

= \frac{sin ( y ) - e ^{c o s (y)}}{- 1}

Apply the fraction rule:

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

= - \frac{sin ( y ) - e ^{c o s (y)}}{1}

Apply rule

\frac{a}{1} = a

= - (sin (y) - e^{c o s (y)})

Distribute parentheses

= - sin (y) - (- e^{c o s (y)})

Apply minus-plus rules

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

= - sin (y) + e^{c o s (y)}

e^{s i n (x)} = - sin (y) + e^{c o s (y)}

e^{s i n (x)} = - sin (y) + e^{c o s (y)}

e^{s i n (x)} = - sin (y) + e^{c o s (y)}

Apply exponent rules

e^{s i n (x)} = - sin (y) + e^{c o s (y)}

If

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{s i n (x)}) = ln (- sin (y) + e^{c o s (y)})

Apply log rule:

ln (e^{a}) = a

ln (e^{s i n (x)}) = sin (x)

sin (x) = ln (- sin (y) + e^{c o s (y)})

sin (x) = ln (- sin (y) + e^{c o s (y)})

Apply trig inverse properties

sin (x) = ln (- sin (y) + e^{c o s (y)})

General solutions for

sin (x) = ln (- sin (y) + e^{c o s (y)})

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (ln (- sin (y) + e^{c o s (y)})) + 2 πn, x = π + arcsin (- ln (- sin (y) + e^{c o s (y)})) + 2 πn

x = arcsin (ln (- sin (y) + e^{c o s (y)})) + 2 πn, x = π + arcsin (- ln (- sin (y) + e^{c o s (y)})) + 2 πn

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