What is the general solution for 120=23sin(pi/6 (x))+107 ?

The general solution for 120=23sin(pi/6 (x))+107 is x=(6*0.60069…)/pi+12n,x=6-(6*0.60069…)/pi+12n

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120=23sin(pi/6 (x))+107

Solution

120 = 23 sin (\frac{π}{6} (x)) + 107

Solution

x = \frac{6 \cdot 0.60069 \dots}{π} + 12 n, x = 6 - \frac{6 \cdot 0.60069 \dots}{π} + 12 n

Degrees

x = 65.73237 \dots^{\circ} + 687.54935 \dots^{\circ} n, x = 278.04230 \dots^{\circ} + 687.54935 \dots^{\circ} n

Solution steps

120 = 23 sin (\frac{π}{6} (x)) + 107

Switch sides

23 sin (\frac{π}{6} x) + 107 = 120

Move

107

to the right side

23 sin (\frac{π}{6} x) + 107 = 120

Subtract

107

from both sides

23 sin (\frac{π}{6} x) + 107 - 107 = 120 - 107

Simplify

23 sin (\frac{π}{6} x) = 13

23 sin (\frac{π}{6} x) = 13

Divide both sides by

23

23 sin (\frac{π}{6} x) = 13

Divide both sides by

23

\frac{23 sin ( \frac{π}{6} x )}{23} = \frac{13}{23}

Simplify

sin (\frac{π}{6} x) = \frac{13}{23}

sin (\frac{π}{6} x) = \frac{13}{23}

Apply trig inverse properties

sin (\frac{π}{6} x) = \frac{13}{23}

General solutions for

sin (\frac{π}{6} x) = \frac{13}{23}

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

\frac{π}{6} x = arcsin (\frac{13}{23}) + 2 πn, \frac{π}{6} x = π - arcsin (\frac{13}{23}) + 2 πn

\frac{π}{6} x = arcsin (\frac{13}{23}) + 2 πn, \frac{π}{6} x = π - arcsin (\frac{13}{23}) + 2 πn

Solve

\frac{π}{6} x = arcsin (\frac{13}{23}) + 2 πn : x = \frac{6 arcsin ( \frac{13}{23} )}{π} + 12 n

\frac{π}{6} x = arcsin (\frac{13}{23}) + 2 πn

Multiply both sides by

6

\frac{π}{6} x = arcsin (\frac{13}{23}) + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} x = 6 arcsin (\frac{13}{23}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} x = 6 arcsin (\frac{13}{23}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} x : π x

6 \cdot \frac{π}{6} x

Multiply fractions:

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

= \frac{6 π}{6} x

Cancel the common factor:

6

= x π

Simplify

6 arcsin (\frac{13}{23}) + 6 \cdot 2 πn : 6 arcsin (\frac{13}{23}) + 12 πn

6 arcsin (\frac{13}{23}) + 6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 6 arcsin (\frac{13}{23}) + 12 πn

π x = 6 arcsin (\frac{13}{23}) + 12 πn

π x = 6 arcsin (\frac{13}{23}) + 12 πn

π x = 6 arcsin (\frac{13}{23}) + 12 πn

Divide both sides by

π

π x = 6 arcsin (\frac{13}{23}) + 12 πn

Divide both sides by

π

\frac{π x}{π} = \frac{6 arcsin ( \frac{13}{23} )}{π} + \frac{12 πn}{π}

Simplify

x = \frac{6 arcsin ( \frac{13}{23} )}{π} + 12 n

x = \frac{6 arcsin ( \frac{13}{23} )}{π} + 12 n

Solve

\frac{π}{6} x = π - arcsin (\frac{13}{23}) + 2 πn : x = 6 - \frac{6 arcsin ( \frac{13}{23} )}{π} + 12 n

\frac{π}{6} x = π - arcsin (\frac{13}{23}) + 2 πn

Multiply both sides by

6

\frac{π}{6} x = π - arcsin (\frac{13}{23}) + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} x = 6 π - 6 arcsin (\frac{13}{23}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} x = 6 π - 6 arcsin (\frac{13}{23}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} x : π x

6 \cdot \frac{π}{6} x

Multiply fractions:

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

= \frac{6 π}{6} x

Cancel the common factor:

6

= x π

Simplify

6 π - 6 arcsin (\frac{13}{23}) + 6 \cdot 2 πn : 6 π - 6 arcsin (\frac{13}{23}) + 12 πn

6 π - 6 arcsin (\frac{13}{23}) + 6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 6 π - 6 arcsin (\frac{13}{23}) + 12 πn

π x = 6 π - 6 arcsin (\frac{13}{23}) + 12 πn

π x = 6 π - 6 arcsin (\frac{13}{23}) + 12 πn

π x = 6 π - 6 arcsin (\frac{13}{23}) + 12 πn

Divide both sides by

π

π x = 6 π - 6 arcsin (\frac{13}{23}) + 12 πn

Divide both sides by

π

\frac{π x}{π} = \frac{6 π}{π} - \frac{6 arcsin ( \frac{13}{23} )}{π} + \frac{12 πn}{π}

Simplify

\frac{π x}{π} = \frac{6 π}{π} - \frac{6 arcsin ( \frac{13}{23} )}{π} + \frac{12 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{6 π}{π} - \frac{6 arcsin ( \frac{13}{23} )}{π} + \frac{12 πn}{π} : 6 - \frac{6 arcsin ( \frac{13}{23} )}{π} + 12 n

\frac{6 π}{π} - \frac{6 arcsin ( \frac{13}{23} )}{π} + \frac{12 πn}{π}

Cancel

\frac{6 π}{π} : 6

\frac{6 π}{π}

Cancel the common factor:

π

= 6

= 6 - \frac{6 arcsin ( \frac{13}{23} )}{π} + \frac{12 πn}{π}

Cancel

\frac{12 πn}{π} : 12 n

\frac{12 πn}{π}

Cancel the common factor:

π

= 12 n

= 6 - \frac{6 arcsin ( \frac{13}{23} )}{π} + 12 n

x = 6 - \frac{6 arcsin ( \frac{13}{23} )}{π} + 12 n

x = 6 - \frac{6 arcsin ( \frac{13}{23} )}{π} + 12 n

x = 6 - \frac{6 arcsin ( \frac{13}{23} )}{π} + 12 n

x = \frac{6 arcsin ( \frac{13}{23} )}{π} + 12 n, x = 6 - \frac{6 arcsin ( \frac{13}{23} )}{π} + 12 n

Show solutions in decimal form

x = \frac{6 \cdot 0.60069 \dots}{π} + 12 n, x = 6 - \frac{6 \cdot 0.60069 \dots}{π} + 12 n

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Frequently Asked Questions (FAQ)

What is the general solution for 120=23sin(pi/6 (x))+107 ?
The general solution for 120=23sin(pi/6 (x))+107 is x=(6*0.60069…)/pi+12n,x=6-(6*0.60069…)/pi+12n