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Popular Trigonometry >

solvefor x,(sin(pix)+cos(piy))^5=36

Solution

solve for

x, (sin (π x) + cos (π y))^{5} = 36

Solution

x = \frac{arcsin ( 5 36 - cos ( π y ) )}{π} + 2 n, x = 1 + \frac{arcsin ( - 5 36 + cos ( π y ) )}{π} + 2 n

Solution steps

(sin (π x) + cos (π y))^{5} = 36

For

(g (x))^{n} = f (a)

, n is odd, the solution is

g (x) = n f (a)

Solve

sin (π x) + cos (π y) = 536 : sin (π x) = 536 - cos (π y)

sin (π x) + cos (π y) = 536

Move

cos (π y)

to the right side

sin (π x) + cos (π y) = 536

Subtract

cos (π y)

from both sides

sin (π x) + cos (π y) - cos (π y) = 536 - cos (π y)

Simplify

sin (π x) = 536 - cos (π y)

sin (π x) = 536 - cos (π y)

The solution is

sin (π x) = 536 - cos (π y)

Apply trig inverse properties

sin (π x) = 536 - cos (π y)

General solutions for

sin (π x) = 536 - cos (π y)

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

π x = arcsin (536 - cos (π y)) + 2 πn, π x = π + arcsin (- 536 + cos (π y)) + 2 πn

π x = arcsin (536 - cos (π y)) + 2 πn, π x = π + arcsin (- 536 + cos (π y)) + 2 πn

Solve

π x = arcsin (536 - cos (π y)) + 2 πn : x = \frac{arcsin ( 5 36 - cos ( π y ) )}{π} + 2 n

π x = arcsin (536 - cos (π y)) + 2 πn

Divide both sides by

π

π x = arcsin (536 - cos (π y)) + 2 πn

Divide both sides by

π

\frac{π x}{π} = \frac{arcsin ( 5 36 - cos ( π y ) )}{π} + \frac{2 πn}{π}

Simplify

x = \frac{arcsin ( 5 36 - cos ( π y ) )}{π} + 2 n

x = \frac{arcsin ( 5 36 - cos ( π y ) )}{π} + 2 n

Solve

π x = π + arcsin (- 536 + cos (π y)) + 2 πn : x = 1 + \frac{arcsin ( - 5 36 + cos ( π y ) )}{π} + 2 n

π x = π + arcsin (- 536 + cos (π y)) + 2 πn

Divide both sides by

π

π x = π + arcsin (- 536 + cos (π y)) + 2 πn

Divide both sides by

π

\frac{π x}{π} = \frac{π}{π} + \frac{arcsin ( - 5 36 + cos ( π y ) )}{π} + \frac{2 πn}{π}

Simplify

\frac{π x}{π} = \frac{π}{π} + \frac{arcsin ( - 5 36 + cos ( π y ) )}{π} + \frac{2 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{π}{π} + \frac{arcsin ( - 5 36 + cos ( π y ) )}{π} + \frac{2 πn}{π} : 1 + \frac{arcsin ( - 5 36 + cos ( π y ) )}{π} + 2 n

\frac{π}{π} + \frac{arcsin ( - 5 36 + cos ( π y ) )}{π} + \frac{2 πn}{π}

Apply rule

\frac{a}{a} = 1

\frac{π}{π} = 1

= 1 + \frac{arcsin ( cos ( π y ) - 5 36 )}{π} + \frac{2 πn}{π}

Cancel

\frac{2 πn}{π} : 2 n

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

= 1 + \frac{arcsin ( cos ( π y ) - 5 36 )}{π} + 2 n

x = 1 + \frac{arcsin ( - 5 36 + cos ( π y ) )}{π} + 2 n

x = 1 + \frac{arcsin ( - 5 36 + cos ( π y ) )}{π} + 2 n

x = 1 + \frac{arcsin ( - 5 36 + cos ( π y ) )}{π} + 2 n

x = \frac{arcsin ( 5 36 - cos ( π y ) )}{π} + 2 n, x = 1 + \frac{arcsin ( - 5 36 + cos ( π y ) )}{π} + 2 n

Graph

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