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Popular Trigonometry >

solvefor n,y=-sin(2((3pi)/4+pin))2

Solution

solvefor

Solution

n = \frac{arcsin ( - \frac{y}{2} )}{2 π} + k - \frac{3}{4}, n = k + \frac{- π + 2 arcsin ( \frac{y}{2} )}{4 π}

Solution steps

y = - sin (2 (\frac{3 π}{4} + πn)) \cdot 2

Switch sides

- sin (2 (\frac{3 π}{4} + πn)) \cdot 2 = y

Divide both sides by

- 2

- sin (2 (\frac{3 π}{4} + πn)) \cdot 2 = y

Divide both sides by

- 2

\frac{- sin ( 2 ( \frac{3 π}{4} + πn ) ) \cdot 2}{- 2} = \frac{y}{- 2}

Simplify

sin (2 (\frac{3 π}{4} + πn)) = - \frac{y}{2}

sin (2 (\frac{3 π}{4} + πn)) = - \frac{y}{2}

Apply trig inverse properties

sin (2 (\frac{3 π}{4} + πn)) = - \frac{y}{2}

General solutions for

sin (2 (\frac{3 π}{4} + πn)) = - \frac{y}{2}

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

2 (\frac{3 π}{4} + πn) = arcsin (- \frac{y}{2}) + 2 πk, 2 (\frac{3 π}{4} + πn) = π + arcsin (\frac{y}{2}) + 2 πk

2 (\frac{3 π}{4} + πn) = arcsin (- \frac{y}{2}) + 2 πk, 2 (\frac{3 π}{4} + πn) = π + arcsin (\frac{y}{2}) + 2 πk

Solve

2 (\frac{3 π}{4} + πn) = arcsin (- \frac{y}{2}) + 2 πk : n = \frac{arcsin ( - \frac{y}{2} )}{2 π} + k - \frac{3}{4}

2 (\frac{3 π}{4} + πn) = arcsin (- \frac{y}{2}) + 2 πk

Divide both sides by

2

2 (\frac{3 π}{4} + πn) = arcsin (- \frac{y}{2}) + 2 πk

Divide both sides by

2

\frac{2 ( \frac{3 π}{4} + πn )}{2} = \frac{arcsin ( - \frac{y}{2} )}{2} + \frac{2 πk}{2}

Simplify

\frac{3 π}{4} + πn = \frac{arcsin ( - \frac{y}{2} )}{2} + πk

\frac{3 π}{4} + πn = \frac{arcsin ( - \frac{y}{2} )}{2} + πk

Move

\frac{3 π}{4}

to the right side

\frac{3 π}{4} + πn = \frac{arcsin ( - \frac{y}{2} )}{2} + πk

Subtract

\frac{3 π}{4}

from both sides

\frac{3 π}{4} + πn - \frac{3 π}{4} = \frac{arcsin ( - \frac{y}{2} )}{2} + πk - \frac{3 π}{4}

Simplify

πn = \frac{arcsin ( - \frac{y}{2} )}{2} + πk - \frac{3 π}{4}

πn = \frac{arcsin ( - \frac{y}{2} )}{2} + πk - \frac{3 π}{4}

Divide both sides by

π

πn = \frac{arcsin ( - \frac{y}{2} )}{2} + πk - \frac{3 π}{4}

Divide both sides by

π

\frac{πn}{π} = \frac{\frac{a r c s i n ( - \frac{y}{2} )}{2}}{π} + \frac{πk}{π} - \frac{\frac{3 π}{4}}{π}

Simplify

\frac{πn}{π} = \frac{\frac{a r c s i n ( - \frac{y}{2} )}{2}}{π} + \frac{πk}{π} - \frac{\frac{3 π}{4}}{π}

Simplify

\frac{πn}{π} : n

\frac{πn}{π}

Cancel the common factor:

π

= n

Simplify

\frac{\frac{a r c s i n ( - \frac{y}{2} )}{2}}{π} + \frac{πk}{π} - \frac{\frac{3 π}{4}}{π} : \frac{arcsin ( - \frac{y}{2} )}{2 π} + k - \frac{3}{4}

\frac{\frac{a r c s i n ( - \frac{y}{2} )}{2}}{π} + \frac{πk}{π} - \frac{\frac{3 π}{4}}{π}

\frac{\frac{a r c s i n ( - \frac{y}{2} )}{2}}{π} = \frac{arcsin ( - \frac{y}{2} )}{2 π}

\frac{\frac{a r c s i n ( - \frac{y}{2} )}{2}}{π}

Apply the fraction rule:

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

= \frac{arcsin ( - \frac{y}{2} )}{2 π}

\frac{πk}{π} = k

\frac{πk}{π}

Cancel the common factor:

π

= k

\frac{\frac{3 π}{4}}{π} = \frac{3}{4}

\frac{\frac{3 π}{4}}{π}

Apply the fraction rule:

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

= \frac{3 π}{4 π}

Cancel the common factor:

π

= \frac{3}{4}

= \frac{arcsin ( - \frac{y}{2} )}{2 π} + k - \frac{3}{4}

n = \frac{arcsin ( - \frac{y}{2} )}{2 π} + k - \frac{3}{4}

n = \frac{arcsin ( - \frac{y}{2} )}{2 π} + k - \frac{3}{4}

n = \frac{arcsin ( - \frac{y}{2} )}{2 π} + k - \frac{3}{4}

Solve

2 (\frac{3 π}{4} + πn) = π + arcsin (\frac{y}{2}) + 2 πk : n = k + \frac{- π + 2 arcsin ( \frac{y}{2} )}{4 π}

2 (\frac{3 π}{4} + πn) = π + arcsin (\frac{y}{2}) + 2 πk

Divide both sides by

2

2 (\frac{3 π}{4} + πn) = π + arcsin (\frac{y}{2}) + 2 πk

Divide both sides by

2

\frac{2 ( \frac{3 π}{4} + πn )}{2} = \frac{π}{2} + \frac{arcsin ( \frac{y}{2} )}{2} + \frac{2 πk}{2}

Simplify

\frac{3 π}{4} + πn = \frac{π}{2} + \frac{arcsin ( \frac{y}{2} )}{2} + πk

\frac{3 π}{4} + πn = \frac{π}{2} + \frac{arcsin ( \frac{y}{2} )}{2} + πk

Move

\frac{3 π}{4}

to the right side

\frac{3 π}{4} + πn = \frac{π}{2} + \frac{arcsin ( \frac{y}{2} )}{2} + πk

Subtract

\frac{3 π}{4}

from both sides

\frac{3 π}{4} + πn - \frac{3 π}{4} = \frac{π}{2} + \frac{arcsin ( \frac{y}{2} )}{2} + πk - \frac{3 π}{4}

Simplify

\frac{3 π}{4} + πn - \frac{3 π}{4} = \frac{π}{2} + \frac{arcsin ( \frac{y}{2} )}{2} + πk - \frac{3 π}{4}

Simplify

\frac{3 π}{4} + πn - \frac{3 π}{4} : πn

\frac{3 π}{4} + πn - \frac{3 π}{4}

Add similar elements:

\frac{3 π}{4} - \frac{3 π}{4} = 0

= πn

Simplify

\frac{π}{2} + \frac{arcsin ( \frac{y}{2} )}{2} + πk - \frac{3 π}{4} : πk + \frac{- π + 2 arcsin ( \frac{y}{2} )}{4}

\frac{π}{2} + \frac{arcsin ( \frac{y}{2} )}{2} + πk - \frac{3 π}{4}

Group like terms

= πk + \frac{π}{2} - \frac{3 π}{4} + \frac{arcsin ( \frac{y}{2} )}{2}

Combine the fractions

\frac{π}{2} + \frac{arcsin ( \frac{y}{2} )}{2} : \frac{π + arcsin ( \frac{y}{2} )}{2}

Apply rule

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

= \frac{π + arcsin ( \frac{y}{2} )}{2}

= πk + \frac{arcsin ( \frac{y}{2} ) + π}{2} - \frac{3 π}{4}

Least Common Multiplier of

2, 4 : 4

2, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

2

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

4

For

\frac{π + arcsin ( \frac{y}{2} )}{2} :

multiply the denominator and numerator by

2

\frac{π + arcsin ( \frac{y}{2} )}{2} = \frac{( π + arcsin ( \frac{y}{2} ) ) \cdot 2}{2 \cdot 2} = \frac{( π + arcsin ( \frac{y}{2} ) ) \cdot 2}{4}

= \frac{( π + arcsin ( \frac{y}{2} ) ) \cdot 2}{4} - \frac{3 π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{( π + arcsin ( \frac{y}{2} ) ) \cdot 2 - 3 π}{4}

Expand

(π + arcsin (\frac{y}{2})) \cdot 2 - 3 π : - π + 2 arcsin (\frac{y}{2})

(π + arcsin (\frac{y}{2})) \cdot 2 - 3 π

= 2 (π + arcsin (\frac{y}{2})) - 3 π

Expand

2 (π + arcsin (\frac{y}{2})) : 2 π + 2 arcsin (\frac{y}{2})

2 (π + arcsin (\frac{y}{2}))

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = π, c = arcsin (\frac{y}{2})

= 2 π + 2 arcsin (\frac{y}{2})

= 2 π + 2 arcsin (\frac{y}{2}) - 3 π

Simplify

2 π + 2 arcsin (\frac{y}{2}) - 3 π : - π + 2 arcsin (\frac{y}{2})

2 π + 2 arcsin (\frac{y}{2}) - 3 π

Group like terms

= 2 π - 3 π + 2 arcsin (\frac{y}{2})

Add similar elements:

2 π - 3 π = - π

= - π + 2 arcsin (\frac{y}{2})

= - π + 2 arcsin (\frac{y}{2})

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

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

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

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

Divide both sides by

π

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

Divide both sides by

π

\frac{πn}{π} = \frac{πk}{π} + \frac{\frac{- π + 2 a r c s i n ( \frac{y}{2} )}{4}}{π}

Simplify

\frac{πn}{π} = \frac{πk}{π} + \frac{\frac{- π + 2 a r c s i n ( \frac{y}{2} )}{4}}{π}

Simplify

\frac{πn}{π} : n

\frac{πn}{π}

Cancel the common factor:

π

= n

Simplify

\frac{πk}{π} + \frac{\frac{- π + 2 a r c s i n ( \frac{y}{2} )}{4}}{π} : k + \frac{- π + 2 arcsin ( \frac{y}{2} )}{4 π}

\frac{πk}{π} + \frac{\frac{- π + 2 a r c s i n ( \frac{y}{2} )}{4}}{π}

Cancel

\frac{πk}{π} : k

\frac{πk}{π}

Cancel the common factor:

π

= k

= k + \frac{\frac{2 a r c s i n ( \frac{y}{2} ) - π}{4}}{π}

Apply the fraction rule:

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

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

n = k + \frac{- π + 2 arcsin ( \frac{y}{2} )}{4 π}

n = k + \frac{- π + 2 arcsin ( \frac{y}{2} )}{4 π}

n = k + \frac{- π + 2 arcsin ( \frac{y}{2} )}{4 π}

n = \frac{arcsin ( - \frac{y}{2} )}{2 π} + k - \frac{3}{4}, n = k + \frac{- π + 2 arcsin ( \frac{y}{2} )}{4 π}

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