What is the general solution for 14=-3.75cos(pi/6 x)+11.25 ?

The general solution for 14=-3.75cos(pi/6 x)+11.25 is x=(6*2.39400…)/pi+12n,x=-(6*2.39400…)/pi+12n

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14=-3.75cos(pi/6 x)+11.25

Solution

14 = - 3.75 cos (\frac{π}{6} x) + 11.25

Solution

x = \frac{6 \cdot 2.39400 \dots}{π} + 12 n, x = - \frac{6 \cdot 2.39400 \dots}{π} + 12 n

Degrees

x = 261.96885 \dots^{\circ} + 687.54935 \dots^{\circ} n, x = - 261.96885 \dots^{\circ} + 687.54935 \dots^{\circ} n

Solution steps

14 = - 3.75 cos (\frac{π}{6} x) + 11.25

Switch sides

- 3.75 cos (\frac{π}{6} x) + 11.25 = 14

Multiply both sides by

100

- 3.75 cos (\frac{π}{6} x) + 11.25 = 14

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

- 3.75 cos (\frac{π}{6} x) \cdot 100 + 11.25 \cdot 100 = 14 \cdot 100

Refine

- 375 cos (\frac{π}{6} x) + 1125 = 1400

- 375 cos (\frac{π}{6} x) + 1125 = 1400

Move

1125

to the right side

- 375 cos (\frac{π}{6} x) + 1125 = 1400

Subtract

1125

from both sides

- 375 cos (\frac{π}{6} x) + 1125 - 1125 = 1400 - 1125

Simplify

- 375 cos (\frac{π}{6} x) = 275

- 375 cos (\frac{π}{6} x) = 275

Divide both sides by

- 375

- 375 cos (\frac{π}{6} x) = 275

Divide both sides by

- 375

\frac{- 375 cos ( \frac{π}{6} x )}{- 375} = \frac{275}{- 375}

Simplify

\frac{- 375 cos ( \frac{π}{6} x )}{- 375} = \frac{275}{- 375}

Simplify

\frac{- 375 cos ( \frac{π}{6} x )}{- 375} : cos (\frac{π}{6} x)

\frac{- 375 cos ( \frac{π}{6} x )}{- 375}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{375 cos ( \frac{π}{6} x )}{375}

Divide the numbers:

\frac{375}{375} = 1

= cos (\frac{π}{6} x)

Simplify

\frac{275}{- 375} : - \frac{11}{15}

\frac{275}{- 375}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{275}{375}

Cancel the common factor:

25

= - \frac{11}{15}

cos (\frac{π}{6} x) = - \frac{11}{15}

cos (\frac{π}{6} x) = - \frac{11}{15}

cos (\frac{π}{6} x) = - \frac{11}{15}

Apply trig inverse properties

cos (\frac{π}{6} x) = - \frac{11}{15}

General solutions for

cos (\frac{π}{6} x) = - \frac{11}{15}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

\frac{π}{6} x = arccos (- \frac{11}{15}) + 2 πn, \frac{π}{6} x = - arccos (- \frac{11}{15}) + 2 πn

\frac{π}{6} x = arccos (- \frac{11}{15}) + 2 πn, \frac{π}{6} x = - arccos (- \frac{11}{15}) + 2 πn

Solve

\frac{π}{6} x = arccos (- \frac{11}{15}) + 2 πn : x = \frac{6 arccos ( - \frac{11}{15} )}{π} + 12 n

\frac{π}{6} x = arccos (- \frac{11}{15}) + 2 πn

Multiply both sides by

6

\frac{π}{6} x = arccos (- \frac{11}{15}) + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} x = 6 arccos (- \frac{11}{15}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} x = 6 arccos (- \frac{11}{15}) + 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 arccos (- \frac{11}{15}) + 6 \cdot 2 πn : 6 arccos (- \frac{11}{15}) + 12 πn

6 arccos (- \frac{11}{15}) + 6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 6 arccos (- \frac{11}{15}) + 12 πn

π x = 6 arccos (- \frac{11}{15}) + 12 πn

π x = 6 arccos (- \frac{11}{15}) + 12 πn

π x = 6 arccos (- \frac{11}{15}) + 12 πn

Divide both sides by

π

π x = 6 arccos (- \frac{11}{15}) + 12 πn

Divide both sides by

π

\frac{π x}{π} = \frac{6 arccos ( - \frac{11}{15} )}{π} + \frac{12 πn}{π}

Simplify

x = \frac{6 arccos ( - \frac{11}{15} )}{π} + 12 n

x = \frac{6 arccos ( - \frac{11}{15} )}{π} + 12 n

Solve

\frac{π}{6} x = - arccos (- \frac{11}{15}) + 2 πn : x = - \frac{6 arccos ( - \frac{11}{15} )}{π} + 12 n

\frac{π}{6} x = - arccos (- \frac{11}{15}) + 2 πn

Multiply both sides by

6

\frac{π}{6} x = - arccos (- \frac{11}{15}) + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} x = - 6 arccos (- \frac{11}{15}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} x = - 6 arccos (- \frac{11}{15}) + 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 arccos (- \frac{11}{15}) + 6 \cdot 2 πn : - 6 arccos (- \frac{11}{15}) + 12 πn

- 6 arccos (- \frac{11}{15}) + 6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= - 6 arccos (- \frac{11}{15}) + 12 πn

π x = - 6 arccos (- \frac{11}{15}) + 12 πn

π x = - 6 arccos (- \frac{11}{15}) + 12 πn

π x = - 6 arccos (- \frac{11}{15}) + 12 πn

Divide both sides by

π

π x = - 6 arccos (- \frac{11}{15}) + 12 πn

Divide both sides by

π

\frac{π x}{π} = - \frac{6 arccos ( - \frac{11}{15} )}{π} + \frac{12 πn}{π}

Simplify

x = - \frac{6 arccos ( - \frac{11}{15} )}{π} + 12 n

x = - \frac{6 arccos ( - \frac{11}{15} )}{π} + 12 n

x = \frac{6 arccos ( - \frac{11}{15} )}{π} + 12 n, x = - \frac{6 arccos ( - \frac{11}{15} )}{π} + 12 n

Show solutions in decimal form

x = \frac{6 \cdot 2.39400 \dots}{π} + 12 n, x = - \frac{6 \cdot 2.39400 \dots}{π} + 12 n

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Frequently Asked Questions (FAQ)

What is the general solution for 14=-3.75cos(pi/6 x)+11.25 ?
The general solution for 14=-3.75cos(pi/6 x)+11.25 is x=(6*2.39400…)/pi+12n,x=-(6*2.39400…)/pi+12n