What is the general solution for 75= 120/4*cos((2*pi*x)/(360))+120/2 ?

The general solution for 75= 120/4*cos((2*pi*x)/(360))+120/2 is x=60+360n,x=300+360n

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75= 120/4cos((2pi*x)/(360))+120/2

Solution

75 = \frac{120}{4} \cdot cos (\frac{2 \cdot π \cdot x}{360}) + \frac{120}{2}

Solution

x = 60 + 360 n, x = 300 + 360 n

Degrees

x = 3437.74677 \dots^{\circ} + 20626.48062 \dots^{\circ} n, x = 17188.73385 \dots^{\circ} + 20626.48062 \dots^{\circ} n

Solution steps

75 = \frac{120}{4} cos (\frac{2 π x}{360}) + \frac{120}{2}

Switch sides

\frac{120}{4} cos (\frac{2 π x}{360}) + \frac{120}{2} = 75

\frac{120}{4} cos (\frac{2 π x}{360}) = 30 cos (\frac{2 π x}{360})

\frac{120}{4} cos (\frac{2 π x}{360})

Multiply fractions:

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

= \frac{120 cos ( \frac{2 π x}{360} )}{4}

Divide the numbers:

\frac{120}{4} = 30

= 30 cos (\frac{2 π x}{360})

\frac{120}{2} = 60

\frac{120}{2}

Divide the numbers:

\frac{120}{2} = 60

= 60

30 cos (\frac{2 π x}{360}) + 60 = 75

Move

60

to the right side

30 cos (\frac{2 π x}{360}) + 60 = 75

Subtract

60

from both sides

30 cos (\frac{2 π x}{360}) + 60 - 60 = 75 - 60

Simplify

30 cos (\frac{2 π x}{360}) = 15

30 cos (\frac{2 π x}{360}) = 15

Divide both sides by

30

30 cos (\frac{2 π x}{360}) = 15

Divide both sides by

30

\frac{30 cos ( \frac{2 π x}{360} )}{30} = \frac{15}{30}

Simplify

cos (\frac{2 π x}{360}) = \frac{1}{2}

cos (\frac{2 π x}{360}) = \frac{1}{2}

General solutions for

cos (\frac{2 π x}{360}) = \frac{1}{2}

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

\frac{2 π x}{360} = \frac{π}{3} + 2 πn, \frac{2 π x}{360} = \frac{5 π}{3} + 2 πn

\frac{2 π x}{360} = \frac{π}{3} + 2 πn, \frac{2 π x}{360} = \frac{5 π}{3} + 2 πn

Solve

\frac{2 π x}{360} = \frac{π}{3} + 2 πn : x = 60 + 360 n

\frac{2 π x}{360} = \frac{π}{3} + 2 πn

Multiply both sides by

360

\frac{2 π x}{360} = \frac{π}{3} + 2 πn

Multiply both sides by

360

\frac{360 \cdot 2 π x}{360} = 360 \cdot \frac{π}{3} + 360 \cdot 2 πn

Simplify

\frac{360 \cdot 2 π x}{360} = 360 \cdot \frac{π}{3} + 360 \cdot 2 πn

Simplify

\frac{360 \cdot 2 π x}{360} : 2 π x

\frac{360 \cdot 2 π x}{360}

Multiply the numbers:

360 \cdot 2 = 720

= \frac{720 π x}{360}

Divide the numbers:

\frac{720}{360} = 2

= 2 π x

Simplify

360 \cdot \frac{π}{3} + 360 \cdot 2 πn : 120 π + 720 πn

360 \cdot \frac{π}{3} + 360 \cdot 2 πn

360 \cdot \frac{π}{3} = 120 π

360 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 360}{3}

Divide the numbers:

\frac{360}{3} = 120

= 120 π

360 \cdot 2 πn = 720 πn

360 \cdot 2 πn

Multiply the numbers:

360 \cdot 2 = 720

= 720 πn

= 120 π + 720 πn

2 π x = 120 π + 720 πn

2 π x = 120 π + 720 πn

2 π x = 120 π + 720 πn

Divide both sides by

2 π

2 π x = 120 π + 720 πn

Divide both sides by

2 π

\frac{2 π x}{2 π} = \frac{120 π}{2 π} + \frac{720 πn}{2 π}

Simplify

\frac{2 π x}{2 π} = \frac{120 π}{2 π} + \frac{720 πn}{2 π}

Simplify

\frac{2 π x}{2 π} : x

\frac{2 π x}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{120 π}{2 π} + \frac{720 πn}{2 π} : 60 + 360 n

\frac{120 π}{2 π} + \frac{720 πn}{2 π}

Cancel

\frac{120 π}{2 π} : 60

\frac{120 π}{2 π}

Cancel

\frac{120 π}{2 π} : 60

\frac{120 π}{2 π}

Divide the numbers:

\frac{120}{2} = 60

= \frac{60 π}{π}

Cancel the common factor:

π

= 60

= 60

= 60 + \frac{720 πn}{2 π}

Cancel

\frac{720 πn}{2 π} : 360 n

\frac{720 πn}{2 π}

Cancel

\frac{720 πn}{2 π} : 360 n

\frac{720 πn}{2 π}

Divide the numbers:

\frac{720}{2} = 360

= \frac{360 πn}{π}

Cancel the common factor:

π

= 360 n

= 360 n

= 60 + 360 n

x = 60 + 360 n

x = 60 + 360 n

x = 60 + 360 n

Solve

\frac{2 π x}{360} = \frac{5 π}{3} + 2 πn : x = 300 + 360 n

\frac{2 π x}{360} = \frac{5 π}{3} + 2 πn

Multiply both sides by

360

\frac{2 π x}{360} = \frac{5 π}{3} + 2 πn

Multiply both sides by

360

\frac{360 \cdot 2 π x}{360} = 360 \cdot \frac{5 π}{3} + 360 \cdot 2 πn

Simplify

\frac{360 \cdot 2 π x}{360} = 360 \cdot \frac{5 π}{3} + 360 \cdot 2 πn

Simplify

\frac{360 \cdot 2 π x}{360} : 2 π x

\frac{360 \cdot 2 π x}{360}

Multiply the numbers:

360 \cdot 2 = 720

= \frac{720 π x}{360}

Divide the numbers:

\frac{720}{360} = 2

= 2 π x

Simplify

360 \cdot \frac{5 π}{3} + 360 \cdot 2 πn : 600 π + 720 πn

360 \cdot \frac{5 π}{3} + 360 \cdot 2 πn

360 \cdot \frac{5 π}{3} = 600 π

360 \cdot \frac{5 π}{3}

Multiply fractions:

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

= \frac{5 π 360}{3}

Multiply the numbers:

5 \cdot 360 = 1800

= \frac{1800 π}{3}

Divide the numbers:

\frac{1800}{3} = 600

= 600 π

360 \cdot 2 πn = 720 πn

360 \cdot 2 πn

Multiply the numbers:

360 \cdot 2 = 720

= 720 πn

= 600 π + 720 πn

2 π x = 600 π + 720 πn

2 π x = 600 π + 720 πn

2 π x = 600 π + 720 πn

Divide both sides by

2 π

2 π x = 600 π + 720 πn

Divide both sides by

2 π

\frac{2 π x}{2 π} = \frac{600 π}{2 π} + \frac{720 πn}{2 π}

Simplify

\frac{2 π x}{2 π} = \frac{600 π}{2 π} + \frac{720 πn}{2 π}

Simplify

\frac{2 π x}{2 π} : x

\frac{2 π x}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{600 π}{2 π} + \frac{720 πn}{2 π} : 300 + 360 n

\frac{600 π}{2 π} + \frac{720 πn}{2 π}

Cancel

\frac{600 π}{2 π} : 300

\frac{600 π}{2 π}

Cancel

\frac{600 π}{2 π} : 300

\frac{600 π}{2 π}

Divide the numbers:

\frac{600}{2} = 300

= \frac{300 π}{π}

Cancel the common factor:

π

= 300

= 300

= 300 + \frac{720 πn}{2 π}

Cancel

\frac{720 πn}{2 π} : 360 n

\frac{720 πn}{2 π}

Cancel

\frac{720 πn}{2 π} : 360 n

\frac{720 πn}{2 π}

Divide the numbers:

\frac{720}{2} = 360

= \frac{360 πn}{π}

Cancel the common factor:

π

= 360 n

= 360 n

= 300 + 360 n

x = 300 + 360 n

x = 300 + 360 n

x = 300 + 360 n

x = 60 + 360 n, x = 300 + 360 n

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Frequently Asked Questions (FAQ)

What is the general solution for 75= 120/4*cos((2*pi*x)/(360))+120/2 ?
The general solution for 75= 120/4*cos((2*pi*x)/(360))+120/2 is x=60+360n,x=300+360n