What is the general solution for 30=30cos(1/4 (x-5))+15 ?

The general solution for 30=30cos(1/4 (x-5))+15 is x=(4pi)/3+8pin+5,x=(20pi)/3+8pin+5

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30=30cos(1/4 (x-5))+15

Solution

30 = 30 cos (\frac{1}{4} (x - 5)) + 15

Solution

x = \frac{4 π}{3} + 8 πn + 5, x = \frac{20 π}{3} + 8 πn + 5

Degrees

x = 526.47889 \dots^{\circ} + 144 0^{\circ} n, x = 1486.47889 \dots^{\circ} + 144 0^{\circ} n

Solution steps

30 = 30 cos (\frac{1}{4} (x - 5)) + 15

Switch sides

30 cos (\frac{1}{4} (x - 5)) + 15 = 30

Move

15

to the right side

30 cos (\frac{1}{4} (x - 5)) + 15 = 30

Subtract

15

from both sides

30 cos (\frac{1}{4} (x - 5)) + 15 - 15 = 30 - 15

Simplify

30 cos (\frac{1}{4} (x - 5)) = 15

30 cos (\frac{1}{4} (x - 5)) = 15

Divide both sides by

30

30 cos (\frac{1}{4} (x - 5)) = 15

Divide both sides by

30

\frac{30 cos ( \frac{1}{4} ( x - 5 ) )}{30} = \frac{15}{30}

Simplify

cos (\frac{1}{4} (x - 5)) = \frac{1}{2}

cos (\frac{1}{4} (x - 5)) = \frac{1}{2}

General solutions for

cos (\frac{1}{4} (x - 5)) = \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{1}{4} (x - 5) = \frac{π}{3} + 2 πn, \frac{1}{4} (x - 5) = \frac{5 π}{3} + 2 πn

\frac{1}{4} (x - 5) = \frac{π}{3} + 2 πn, \frac{1}{4} (x - 5) = \frac{5 π}{3} + 2 πn

Solve

\frac{1}{4} (x - 5) = \frac{π}{3} + 2 πn : x = \frac{4 π}{3} + 8 πn + 5

\frac{1}{4} (x - 5) = \frac{π}{3} + 2 πn

Multiply both sides by

4

\frac{1}{4} (x - 5) = \frac{π}{3} + 2 πn

Multiply both sides by

4

4 \cdot \frac{1}{4} (x - 5) = 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Simplify

4 \cdot \frac{1}{4} (x - 5) = 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Simplify

4 \cdot \frac{1}{4} (x - 5) : x - 5

4 \cdot \frac{1}{4} (x - 5)

Multiply fractions:

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

= \frac{1 \cdot 4}{4} (x - 5)

Cancel the common factor:

4

= (x - 5) \cdot 1

Refine

= x - 5

Simplify

4 \cdot \frac{π}{3} + 4 \cdot 2 πn : \frac{4 π}{3} + 8 πn

4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Multiply

4 \cdot \frac{π}{3} : \frac{4 π}{3}

4 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 4}{3}

= \frac{4 π}{3} + 4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= \frac{4 π}{3} + 8 πn

x - 5 = \frac{4 π}{3} + 8 πn

x - 5 = \frac{4 π}{3} + 8 πn

x - 5 = \frac{4 π}{3} + 8 πn

Move

5

to the right side

x - 5 = \frac{4 π}{3} + 8 πn

Add

5

to both sides

x - 5 + 5 = \frac{4 π}{3} + 8 πn + 5

Simplify

x = \frac{4 π}{3} + 8 πn + 5

x = \frac{4 π}{3} + 8 πn + 5

Solve

\frac{1}{4} (x - 5) = \frac{5 π}{3} + 2 πn : x = \frac{20 π}{3} + 8 πn + 5

\frac{1}{4} (x - 5) = \frac{5 π}{3} + 2 πn

Multiply both sides by

4

\frac{1}{4} (x - 5) = \frac{5 π}{3} + 2 πn

Multiply both sides by

4

4 \cdot \frac{1}{4} (x - 5) = 4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

Simplify

4 \cdot \frac{1}{4} (x - 5) = 4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

Simplify

4 \cdot \frac{1}{4} (x - 5) : x - 5

4 \cdot \frac{1}{4} (x - 5)

Multiply fractions:

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

= \frac{1 \cdot 4}{4} (x - 5)

Cancel the common factor:

4

= (x - 5) \cdot 1

Refine

= x - 5

Simplify

4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn : \frac{20 π}{3} + 8 πn

4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

4 \cdot \frac{5 π}{3} = \frac{20 π}{3}

4 \cdot \frac{5 π}{3}

Multiply fractions:

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

= \frac{5 π 4}{3}

Multiply the numbers:

5 \cdot 4 = 20

= \frac{20 π}{3}

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= \frac{20 π}{3} + 8 πn

x - 5 = \frac{20 π}{3} + 8 πn

x - 5 = \frac{20 π}{3} + 8 πn

x - 5 = \frac{20 π}{3} + 8 πn

Move

5

to the right side

x - 5 = \frac{20 π}{3} + 8 πn

Add

5

to both sides

x - 5 + 5 = \frac{20 π}{3} + 8 πn + 5

Simplify

x = \frac{20 π}{3} + 8 πn + 5

x = \frac{20 π}{3} + 8 πn + 5

x = \frac{4 π}{3} + 8 πn + 5, x = \frac{20 π}{3} + 8 πn + 5

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 30=30cos(1/4 (x-5))+15 ?
The general solution for 30=30cos(1/4 (x-5))+15 is x=(4pi)/3+8pin+5,x=(20pi)/3+8pin+5