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solvefor t,y=-y+sin(t)+cos(t)

Solution

solve for

t, y = - y + sin (t) + cos (t)

Solution

t = arcsin (2 y) + 2 πn - \frac{π}{4}, t = π + arcsin (- 2 y) + 2 πn - \frac{π}{4}

Solution steps

y = - y + sin (t) + cos (t)

Switch sides

- y + sin (t) + cos (t) = y

Rewrite using trig identities

- y + sin (t) + cos (t)

sin (t) + cos (t) = 2 sin (t + \frac{π}{4})

sin (t) + cos (t)

Rewrite as

= 2 (\frac{1}{2} sin (t) + \frac{1}{2} cos (t))

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{1}{2}

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (\frac{π}{4}) sin (t) + sin (\frac{π}{4}) cos (t))

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= 2 sin (t + \frac{π}{4})

= - y + 2 sin (t + \frac{π}{4})

- y + 2 sin (t + \frac{π}{4}) = y

Move

y

to the right side

- y + 2 sin (t + \frac{π}{4}) = y

Add

y

to both sides

- y + 2 sin (t + \frac{π}{4}) + y = y + y

Simplify

2 sin (t + \frac{π}{4}) = 2 y

2 sin (t + \frac{π}{4}) = 2 y

Divide both sides by

2

2 sin (t + \frac{π}{4}) = 2 y

Divide both sides by

2

\frac{2 sin ( t + \frac{π}{4} )}{2} = \frac{2 y}{2}

Simplify

\frac{2 sin ( t + \frac{π}{4} )}{2} = \frac{2 y}{2}

Simplify

\frac{2 sin ( t + \frac{π}{4} )}{2} : sin (t + \frac{π}{4})

\frac{2 sin ( t + \frac{π}{4} )}{2}

Cancel the common factor:

2

= sin (t + \frac{π}{4})

Simplify

\frac{2 y}{2} : 2 y

\frac{2 y}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 y}{2 ^{\frac{1}{2}}}

Apply exponent rule:

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

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

= 2^{- \frac{1}{2} + 1} y

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= 2^{\frac{1}{2}} y

Apply radical rule:

2^{\frac{1}{2}} = 2

= 2 y

sin (t + \frac{π}{4}) = 2 y

sin (t + \frac{π}{4}) = 2 y

sin (t + \frac{π}{4}) = 2 y

Apply trig inverse properties

sin (t + \frac{π}{4}) = 2 y

General solutions for

sin (t + \frac{π}{4}) = 2 y

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

t + \frac{π}{4} = arcsin (2 y) + 2 πn, t + \frac{π}{4} = π + arcsin (- 2 y) + 2 πn

t + \frac{π}{4} = arcsin (2 y) + 2 πn, t + \frac{π}{4} = π + arcsin (- 2 y) + 2 πn

Solve

t + \frac{π}{4} = arcsin (2 y) + 2 πn : t = arcsin (2 y) + 2 πn - \frac{π}{4}

t + \frac{π}{4} = arcsin (2 y) + 2 πn

Move

\frac{π}{4}

to the right side

t + \frac{π}{4} = arcsin (2 y) + 2 πn

Subtract

\frac{π}{4}

from both sides

t + \frac{π}{4} - \frac{π}{4} = arcsin (2 y) + 2 πn - \frac{π}{4}

Simplify

t = arcsin (2 y) + 2 πn - \frac{π}{4}

t = arcsin (2 y) + 2 πn - \frac{π}{4}

Solve

t + \frac{π}{4} = π + arcsin (- 2 y) + 2 πn : t = π + arcsin (- 2 y) + 2 πn - \frac{π}{4}

t + \frac{π}{4} = π + arcsin (- 2 y) + 2 πn

Move

\frac{π}{4}

to the right side

t + \frac{π}{4} = π + arcsin (- 2 y) + 2 πn

Subtract

\frac{π}{4}

from both sides

t + \frac{π}{4} - \frac{π}{4} = π + arcsin (- 2 y) + 2 πn - \frac{π}{4}

Simplify

t = π + arcsin (- 2 y) + 2 πn - \frac{π}{4}

t = π + arcsin (- 2 y) + 2 πn - \frac{π}{4}

t = arcsin (2 y) + 2 πn - \frac{π}{4}, t = π + arcsin (- 2 y) + 2 πn - \frac{π}{4}

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