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Popular Trigonometry >

32=37.3sin((2pi)/(365)(x-114))+26

Solution

32 = 37.3 sin (\frac{2 π}{365} (x - 114)) + 26

Solution

x = \frac{365 \cdot 0.16155 \dots}{2 π} + 365 n + 114, x = 365 n + \frac{593}{2} - \frac{365 \cdot 0.16155 \dots}{2 π}

Degrees

x = 7069.45471 \dots^{\circ} + 20912.95952 \dots^{\circ} n, x = 16450.46277 \dots^{\circ} + 20912.95952 \dots^{\circ} n

Solution steps

32 = 37.3 sin (\frac{2 π}{365} (x - 114)) + 26

Switch sides

37.3 sin (\frac{2 π}{365} (x - 114)) + 26 = 32

Multiply both sides by

10

37.3 sin (\frac{2 π}{365} (x - 114)) + 26 = 32

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

37.3 sin (\frac{2 π}{365} (x - 114)) \cdot 10 + 26 \cdot 10 = 32 \cdot 10

Refine

373 sin (\frac{2 π}{365} (x - 114)) + 260 = 320

373 sin (\frac{2 π}{365} (x - 114)) + 260 = 320

Move

260

to the right side

373 sin (\frac{2 π}{365} (x - 114)) + 260 = 320

Subtract

260

from both sides

373 sin (\frac{2 π}{365} (x - 114)) + 260 - 260 = 320 - 260

Simplify

373 sin (\frac{2 π}{365} (x - 114)) = 60

373 sin (\frac{2 π}{365} (x - 114)) = 60

Divide both sides by

373

373 sin (\frac{2 π}{365} (x - 114)) = 60

Divide both sides by

373

\frac{373 sin ( \frac{2 π}{365} ( x - 114 ) )}{373} = \frac{60}{373}

Simplify

sin (\frac{2 π}{365} (x - 114)) = \frac{60}{373}

sin (\frac{2 π}{365} (x - 114)) = \frac{60}{373}

Apply trig inverse properties

sin (\frac{2 π}{365} (x - 114)) = \frac{60}{373}

General solutions for

sin (\frac{2 π}{365} (x - 114)) = \frac{60}{373}

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

\frac{2 π}{365} (x - 114) = arcsin (\frac{60}{373}) + 2 πn, \frac{2 π}{365} (x - 114) = π - arcsin (\frac{60}{373}) + 2 πn

\frac{2 π}{365} (x - 114) = arcsin (\frac{60}{373}) + 2 πn, \frac{2 π}{365} (x - 114) = π - arcsin (\frac{60}{373}) + 2 πn

Solve

\frac{2 π}{365} (x - 114) = arcsin (\frac{60}{373}) + 2 πn : x = \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n + 114

\frac{2 π}{365} (x - 114) = arcsin (\frac{60}{373}) + 2 πn

Multiply both sides by

365

\frac{2 π}{365} (x - 114) = arcsin (\frac{60}{373}) + 2 πn

Multiply both sides by

365

365 \cdot \frac{2 π}{365} (x - 114) = 365 arcsin (\frac{60}{373}) + 365 \cdot 2 πn

Simplify

365 \cdot \frac{2 π}{365} (x - 114) = 365 arcsin (\frac{60}{373}) + 365 \cdot 2 πn

Simplify

365 \cdot \frac{2 π}{365} (x - 114) : 2 π (x - 114)

365 \cdot \frac{2 π}{365} (x - 114)

Multiply fractions:

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

= \frac{2 \cdot 365 π}{365} (x - 114)

Cancel the common factor:

365

= (x - 114) \cdot 2 π

Simplify

365 arcsin (\frac{60}{373}) + 365 \cdot 2 πn : 365 arcsin (\frac{60}{373}) + 730 πn

365 arcsin (\frac{60}{373}) + 365 \cdot 2 πn

Multiply the numbers:

365 \cdot 2 = 730

= 365 arcsin (\frac{60}{373}) + 730 πn

2 π (x - 114) = 365 arcsin (\frac{60}{373}) + 730 πn

2 π (x - 114) = 365 arcsin (\frac{60}{373}) + 730 πn

2 π (x - 114) = 365 arcsin (\frac{60}{373}) + 730 πn

Divide both sides by

2 π

2 π (x - 114) = 365 arcsin (\frac{60}{373}) + 730 πn

Divide both sides by

2 π

\frac{2 π ( x - 114 )}{2 π} = \frac{365 arcsin ( \frac{60}{373} )}{2 π} + \frac{730 πn}{2 π}

Simplify

\frac{2 π ( x - 114 )}{2 π} = \frac{365 arcsin ( \frac{60}{373} )}{2 π} + \frac{730 πn}{2 π}

Simplify

\frac{2 π ( x - 114 )}{2 π} : x - 114

\frac{2 π ( x - 114 )}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π ( x - 114 )}{π}

Cancel the common factor:

π

= x - 114

Simplify

\frac{365 arcsin ( \frac{60}{373} )}{2 π} + \frac{730 πn}{2 π} : \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n

\frac{365 arcsin ( \frac{60}{373} )}{2 π} + \frac{730 πn}{2 π}

Cancel

\frac{730 πn}{2 π} : 365 n

\frac{730 πn}{2 π}

Cancel

\frac{730 πn}{2 π} : 365 n

\frac{730 πn}{2 π}

Divide the numbers:

\frac{730}{2} = 365

= \frac{365 πn}{π}

Cancel the common factor:

π

= 365 n

= 365 n

= \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n

x - 114 = \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n

x - 114 = \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n

x - 114 = \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n

Move

114

to the right side

x - 114 = \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n

Add

114

to both sides

x - 114 + 114 = \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n + 114

Simplify

x = \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n + 114

x = \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n + 114

Solve

\frac{2 π}{365} (x - 114) = π - arcsin (\frac{60}{373}) + 2 πn : x = 365 n + \frac{593}{2} - \frac{365 arcsin ( \frac{60}{373} )}{2 π}

\frac{2 π}{365} (x - 114) = π - arcsin (\frac{60}{373}) + 2 πn

Multiply both sides by

365

\frac{2 π}{365} (x - 114) = π - arcsin (\frac{60}{373}) + 2 πn

Multiply both sides by

365

365 \cdot \frac{2 π}{365} (x - 114) = 365 π - 365 arcsin (\frac{60}{373}) + 365 \cdot 2 πn

Simplify

365 \cdot \frac{2 π}{365} (x - 114) = 365 π - 365 arcsin (\frac{60}{373}) + 365 \cdot 2 πn

Simplify

365 \cdot \frac{2 π}{365} (x - 114) : 2 π (x - 114)

365 \cdot \frac{2 π}{365} (x - 114)

Multiply fractions:

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

= \frac{2 \cdot 365 π}{365} (x - 114)

Cancel the common factor:

365

= (x - 114) \cdot 2 π

Simplify

365 π - 365 arcsin (\frac{60}{373}) + 365 \cdot 2 πn : 365 π - 365 arcsin (\frac{60}{373}) + 730 πn

365 π - 365 arcsin (\frac{60}{373}) + 365 \cdot 2 πn

Multiply the numbers:

365 \cdot 2 = 730

= 365 π - 365 arcsin (\frac{60}{373}) + 730 πn

2 π (x - 114) = 365 π - 365 arcsin (\frac{60}{373}) + 730 πn

2 π (x - 114) = 365 π - 365 arcsin (\frac{60}{373}) + 730 πn

2 π (x - 114) = 365 π - 365 arcsin (\frac{60}{373}) + 730 πn

Divide both sides by

2 π

2 π (x - 114) = 365 π - 365 arcsin (\frac{60}{373}) + 730 πn

Divide both sides by

2 π

\frac{2 π ( x - 114 )}{2 π} = \frac{365 π}{2 π} - \frac{365 arcsin ( \frac{60}{373} )}{2 π} + \frac{730 πn}{2 π}

Simplify

\frac{2 π ( x - 114 )}{2 π} = \frac{365 π}{2 π} - \frac{365 arcsin ( \frac{60}{373} )}{2 π} + \frac{730 πn}{2 π}

Simplify

\frac{2 π ( x - 114 )}{2 π} : x - 114

\frac{2 π ( x - 114 )}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π ( x - 114 )}{π}

Cancel the common factor:

π

= x - 114

Simplify

\frac{365 π}{2 π} - \frac{365 arcsin ( \frac{60}{373} )}{2 π} + \frac{730 πn}{2 π} : \frac{365}{2} - \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n

\frac{365 π}{2 π} - \frac{365 arcsin ( \frac{60}{373} )}{2 π} + \frac{730 πn}{2 π}

Cancel

\frac{365 π}{2 π} : \frac{365}{2}

\frac{365 π}{2 π}

Cancel the common factor:

π

= \frac{365}{2}

= \frac{365}{2} - \frac{365 arcsin ( \frac{60}{373} )}{2 π} + \frac{730 πn}{2 π}

Cancel

\frac{730 πn}{2 π} : 365 n

\frac{730 πn}{2 π}

Cancel

\frac{730 πn}{2 π} : 365 n

\frac{730 πn}{2 π}

Divide the numbers:

\frac{730}{2} = 365

= \frac{365 πn}{π}

Cancel the common factor:

π

= 365 n

= 365 n

= \frac{365}{2} - \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n

x - 114 = \frac{365}{2} - \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n

x - 114 = \frac{365}{2} - \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n

x - 114 = \frac{365}{2} - \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n

Move

114

to the right side

x - 114 = \frac{365}{2} - \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n

Add

114

to both sides

x - 114 + 114 = \frac{365}{2} - \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n + 114

Simplify

x - 114 + 114 = \frac{365}{2} - \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n + 114

Simplify

x - 114 + 114 : x

x - 114 + 114

Add similar elements:

- 114 + 114 = 0

= x

Simplify

\frac{365}{2} - \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n + 114 : 365 n + \frac{593}{2} - \frac{365 arcsin ( \frac{60}{373} )}{2 π}

\frac{365}{2} - \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n + 114

Combine the fractions

114 + \frac{365}{2} : \frac{593}{2}

114 + \frac{365}{2}

Convert element to fraction:

114 = \frac{114 \cdot 2}{2}

= \frac{114 \cdot 2}{2} + \frac{365}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{114 \cdot 2 + 365}{2}

114 \cdot 2 + 365 = 593

114 \cdot 2 + 365

Multiply the numbers:

114 \cdot 2 = 228

= 228 + 365

Add the numbers:

228 + 365 = 593

= 593

= \frac{593}{2}

= 365 n + \frac{593}{2} - \frac{365 arcsin ( \frac{60}{373} )}{2 π}

x = 365 n + \frac{593}{2} - \frac{365 arcsin ( \frac{60}{373} )}{2 π}

x = 365 n + \frac{593}{2} - \frac{365 arcsin ( \frac{60}{373} )}{2 π}

x = 365 n + \frac{593}{2} - \frac{365 arcsin ( \frac{60}{373} )}{2 π}

x = \frac{365 arcsin ( \frac{60}{373} )}{2 π} + 365 n + 114, x = 365 n + \frac{593}{2} - \frac{365 arcsin ( \frac{60}{373} )}{2 π}

Show solutions in decimal form

x = \frac{365 \cdot 0.16155 \dots}{2 π} + 365 n + 114, x = 365 n + \frac{593}{2} - \frac{365 \cdot 0.16155 \dots}{2 π}

Graph

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