What is the general solution for 50=25sin(pi/2 (x+1))+40 ?

The general solution for 50=25sin(pi/2 (x+1))+40 is x=(2*0.41151…)/pi+4n-1,x=4n+1-(2*0.41151…)/pi

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50=25sin(pi/2 (x+1))+40

Solution

50 = 25 sin (\frac{π}{2} (x + 1)) + 40

Solution

x = \frac{2 \cdot 0.41151 \dots}{π} + 4 n - 1, x = 4 n + 1 - \frac{2 \cdot 0.41151 \dots}{π}

Degrees

x = - 42.28544 \dots^{\circ} + 229.18311 \dots^{\circ} n, x = 42.28544 \dots^{\circ} + 229.18311 \dots^{\circ} n

Solution steps

50 = 25 sin (\frac{π}{2} (x + 1)) + 40

Switch sides

25 sin (\frac{π}{2} (x + 1)) + 40 = 50

Move

40

to the right side

25 sin (\frac{π}{2} (x + 1)) + 40 = 50

Subtract

40

from both sides

25 sin (\frac{π}{2} (x + 1)) + 40 - 40 = 50 - 40

Simplify

25 sin (\frac{π}{2} (x + 1)) = 10

25 sin (\frac{π}{2} (x + 1)) = 10

Divide both sides by

25

25 sin (\frac{π}{2} (x + 1)) = 10

Divide both sides by

25

\frac{25 sin ( \frac{π}{2} ( x + 1 ) )}{25} = \frac{10}{25}

Simplify

sin (\frac{π}{2} (x + 1)) = \frac{2}{5}

sin (\frac{π}{2} (x + 1)) = \frac{2}{5}

Apply trig inverse properties

sin (\frac{π}{2} (x + 1)) = \frac{2}{5}

General solutions for

sin (\frac{π}{2} (x + 1)) = \frac{2}{5}

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

\frac{π}{2} (x + 1) = arcsin (\frac{2}{5}) + 2 πn, \frac{π}{2} (x + 1) = π - arcsin (\frac{2}{5}) + 2 πn

\frac{π}{2} (x + 1) = arcsin (\frac{2}{5}) + 2 πn, \frac{π}{2} (x + 1) = π - arcsin (\frac{2}{5}) + 2 πn

Solve

\frac{π}{2} (x + 1) = arcsin (\frac{2}{5}) + 2 πn : x = \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n - 1

\frac{π}{2} (x + 1) = arcsin (\frac{2}{5}) + 2 πn

Multiply both sides by

2

\frac{π}{2} (x + 1) = arcsin (\frac{2}{5}) + 2 πn

Multiply both sides by

2

2 \cdot \frac{π}{2} (x + 1) = 2 arcsin (\frac{2}{5}) + 2 \cdot 2 πn

Simplify

2 \cdot \frac{π}{2} (x + 1) = 2 arcsin (\frac{2}{5}) + 2 \cdot 2 πn

Simplify

2 \cdot \frac{π}{2} (x + 1) : π (x + 1)

2 \cdot \frac{π}{2} (x + 1)

Multiply fractions:

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

= \frac{2 π}{2} (x + 1)

Cancel the common factor:

2

= (x + 1) π

Simplify

2 arcsin (\frac{2}{5}) + 2 \cdot 2 πn : 2 arcsin (\frac{2}{5}) + 4 πn

2 arcsin (\frac{2}{5}) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 arcsin (\frac{2}{5}) + 4 πn

π (x + 1) = 2 arcsin (\frac{2}{5}) + 4 πn

π (x + 1) = 2 arcsin (\frac{2}{5}) + 4 πn

π (x + 1) = 2 arcsin (\frac{2}{5}) + 4 πn

Divide both sides by

π

π (x + 1) = 2 arcsin (\frac{2}{5}) + 4 πn

Divide both sides by

π

\frac{π ( x + 1 )}{π} = \frac{2 arcsin ( \frac{2}{5} )}{π} + \frac{4 πn}{π}

Simplify

x + 1 = \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n

x + 1 = \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n

Move

1

to the right side

x + 1 = \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n

Subtract

1

from both sides

x + 1 - 1 = \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n - 1

Simplify

x = \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n - 1

x = \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n - 1

Solve

\frac{π}{2} (x + 1) = π - arcsin (\frac{2}{5}) + 2 πn : x = 4 n + 1 - \frac{2 arcsin ( \frac{2}{5} )}{π}

\frac{π}{2} (x + 1) = π - arcsin (\frac{2}{5}) + 2 πn

Multiply both sides by

2

\frac{π}{2} (x + 1) = π - arcsin (\frac{2}{5}) + 2 πn

Multiply both sides by

2

2 \cdot \frac{π}{2} (x + 1) = 2 π - 2 arcsin (\frac{2}{5}) + 2 \cdot 2 πn

Simplify

2 \cdot \frac{π}{2} (x + 1) = 2 π - 2 arcsin (\frac{2}{5}) + 2 \cdot 2 πn

Simplify

2 \cdot \frac{π}{2} (x + 1) : π (x + 1)

2 \cdot \frac{π}{2} (x + 1)

Multiply fractions:

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

= \frac{2 π}{2} (x + 1)

Cancel the common factor:

2

= (x + 1) π

Simplify

2 π - 2 arcsin (\frac{2}{5}) + 2 \cdot 2 πn : 2 π - 2 arcsin (\frac{2}{5}) + 4 πn

2 π - 2 arcsin (\frac{2}{5}) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 π - 2 arcsin (\frac{2}{5}) + 4 πn

π (x + 1) = 2 π - 2 arcsin (\frac{2}{5}) + 4 πn

π (x + 1) = 2 π - 2 arcsin (\frac{2}{5}) + 4 πn

π (x + 1) = 2 π - 2 arcsin (\frac{2}{5}) + 4 πn

Divide both sides by

π

π (x + 1) = 2 π - 2 arcsin (\frac{2}{5}) + 4 πn

Divide both sides by

π

\frac{π ( x + 1 )}{π} = \frac{2 π}{π} - \frac{2 arcsin ( \frac{2}{5} )}{π} + \frac{4 πn}{π}

Simplify

\frac{π ( x + 1 )}{π} = \frac{2 π}{π} - \frac{2 arcsin ( \frac{2}{5} )}{π} + \frac{4 πn}{π}

Simplify

\frac{π ( x + 1 )}{π} : x + 1

\frac{π ( x + 1 )}{π}

Cancel the common factor:

π

= x + 1

Simplify

\frac{2 π}{π} - \frac{2 arcsin ( \frac{2}{5} )}{π} + \frac{4 πn}{π} : 2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n

\frac{2 π}{π} - \frac{2 arcsin ( \frac{2}{5} )}{π} + \frac{4 πn}{π}

Cancel

\frac{2 π}{π} : 2

\frac{2 π}{π}

Cancel the common factor:

π

= 2

= 2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + \frac{4 πn}{π}

Cancel

\frac{4 πn}{π} : 4 n

\frac{4 πn}{π}

Cancel the common factor:

π

= 4 n

= 2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n

x + 1 = 2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n

x + 1 = 2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n

x + 1 = 2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n

Move

1

to the right side

x + 1 = 2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n

Subtract

1

from both sides

x + 1 - 1 = 2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n - 1

Simplify

x + 1 - 1 = 2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n - 1

Simplify

x + 1 - 1 : x

x + 1 - 1

Add similar elements:

1 - 1 = 0

= x

Simplify

2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n - 1 : 4 n + 1 - \frac{2 arcsin ( \frac{2}{5} )}{π}

2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n - 1

Subtract the numbers:

2 - 1 = 1

= 4 n + 1 - \frac{2 arcsin ( \frac{2}{5} )}{π}

x = 4 n + 1 - \frac{2 arcsin ( \frac{2}{5} )}{π}

x = 4 n + 1 - \frac{2 arcsin ( \frac{2}{5} )}{π}

x = 4 n + 1 - \frac{2 arcsin ( \frac{2}{5} )}{π}

x = \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n - 1, x = 4 n + 1 - \frac{2 arcsin ( \frac{2}{5} )}{π}

Show solutions in decimal form

x = \frac{2 \cdot 0.41151 \dots}{π} + 4 n - 1, x = 4 n + 1 - \frac{2 \cdot 0.41151 \dots}{π}

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Frequently Asked Questions (FAQ)

What is the general solution for 50=25sin(pi/2 (x+1))+40 ?
The general solution for 50=25sin(pi/2 (x+1))+40 is x=(2*0.41151…)/pi+4n-1,x=4n+1-(2*0.41151…)/pi