What is the general solution for 34sin(pi/6 (x-43))=34 ?

The general solution for 34sin(pi/6 (x-43))=34 is x=12n+46

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34sin(pi/6 (x-43))=34

Solution

34 sin (\frac{π}{6} (x - 43)) = 34

Solution

x = 12 n + 46

Degrees

x = 2635.60585 \dots^{\circ} + 687.54935 \dots^{\circ} n

Solution steps

34 sin (\frac{π}{6} (x - 43)) = 34

Divide both sides by

34

34 sin (\frac{π}{6} (x - 43)) = 34

Divide both sides by

34

\frac{34 sin ( \frac{π}{6} ( x - 43 ) )}{34} = \frac{34}{34}

Simplify

sin (\frac{π}{6} (x - 43)) = 1

sin (\frac{π}{6} (x - 43)) = 1

General solutions for

sin (\frac{π}{6} (x - 43)) = 1

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{π}{6} (x - 43) = \frac{π}{2} + 2 πn

\frac{π}{6} (x - 43) = \frac{π}{2} + 2 πn

Solve

\frac{π}{6} (x - 43) = \frac{π}{2} + 2 πn : x = 12 n + 46

\frac{π}{6} (x - 43) = \frac{π}{2} + 2 πn

Multiply both sides by

6

\frac{π}{6} (x - 43) = \frac{π}{2} + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} (x - 43) = 6 \cdot \frac{π}{2} + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} (x - 43) = 6 \cdot \frac{π}{2} + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} (x - 43) : π (x - 43)

6 \cdot \frac{π}{6} (x - 43)

Multiply fractions:

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

= \frac{6 π}{6} (x - 43)

Cancel the common factor:

6

= (x - 43) π

Simplify

6 \cdot \frac{π}{2} + 6 \cdot 2 πn : 3 π + 12 πn

6 \cdot \frac{π}{2} + 6 \cdot 2 πn

6 \cdot \frac{π}{2} = 3 π

6 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 π

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= 3 π + 12 πn

π (x - 43) = 3 π + 12 πn

π (x - 43) = 3 π + 12 πn

π (x - 43) = 3 π + 12 πn

Divide both sides by

π

π (x - 43) = 3 π + 12 πn

Divide both sides by

π

\frac{π ( x - 43 )}{π} = \frac{3 π}{π} + \frac{12 πn}{π}

Simplify

\frac{π ( x - 43 )}{π} = \frac{3 π}{π} + \frac{12 πn}{π}

Simplify

\frac{π ( x - 43 )}{π} : x - 43

\frac{π ( x - 43 )}{π}

Cancel the common factor:

π

= x - 43

Simplify

\frac{3 π}{π} + \frac{12 πn}{π} : 3 + 12 n

\frac{3 π}{π} + \frac{12 πn}{π}

Cancel

\frac{3 π}{π} : 3

\frac{3 π}{π}

Cancel the common factor:

π

= 3

= 3 + \frac{12 πn}{π}

Cancel

\frac{12 πn}{π} : 12 n

\frac{12 πn}{π}

Cancel the common factor:

π

= 12 n

= 3 + 12 n

x - 43 = 3 + 12 n

x - 43 = 3 + 12 n

x - 43 = 3 + 12 n

Move

43

to the right side

x - 43 = 3 + 12 n

Add

43

to both sides

x - 43 + 43 = 3 + 12 n + 43

Simplify

x = 12 n + 46

x = 12 n + 46

x = 12 n + 46

Popular Examples

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3 sin (3 x) = 2

2sinh(2x)-6cosh(x)=0

2 sinh (2 x) - 6 cosh (x) = 0

-cos(x)=-1

- cos (x) = - 1

sin^2(x)csc(x)=-1/2

sin^{2} (x) csc (x) = - \frac{1}{2}

sec^2(2x)-1=0

sec^{2} (2 x) - 1 = 0

Frequently Asked Questions (FAQ)

What is the general solution for 34sin(pi/6 (x-43))=34 ?
The general solution for 34sin(pi/6 (x-43))=34 is x=12n+46