What is the general solution for 8cos(1/2 x)+4=0 ?

The general solution for 8cos(1/2 x)+4=0 is x=(4pi)/3+4pin,x=(8pi)/3+4pin

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8cos(1/2 x)+4=0

Solution

8 cos (\frac{1}{2} x) + 4 = 0

Solution

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

Degrees

x = 24 0^{\circ} + 72 0^{\circ} n, x = 48 0^{\circ} + 72 0^{\circ} n

Solution steps

8 cos (\frac{1}{2} x) + 4 = 0

Move

4

to the right side

8 cos (\frac{1}{2} x) + 4 = 0

Subtract

4

from both sides

8 cos (\frac{1}{2} x) + 4 - 4 = 0 - 4

Simplify

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

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

Divide both sides by

8

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

Divide both sides by

8

\frac{8 cos ( \frac{1}{2} x )}{8} = \frac{- 4}{8}

Simplify

\frac{8 cos ( \frac{1}{2} x )}{8} = \frac{- 4}{8}

Simplify

\frac{8 cos ( \frac{1}{2} x )}{8} : cos (\frac{1}{2} x)

\frac{8 cos ( \frac{1}{2} x )}{8}

Divide the numbers:

\frac{8}{8} = 1

= cos (\frac{1}{2} x)

Simplify

\frac{- 4}{8} : - \frac{1}{2}

\frac{- 4}{8}

Apply the fraction rule:

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

= - \frac{4}{8}

Cancel the common factor:

4

= - \frac{1}{2}

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

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

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

General solutions for

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

\frac{1}{2} x = \frac{2 π}{3} + 2 πn, \frac{1}{2} x = \frac{4 π}{3} + 2 πn

Solve

\frac{1}{2} x = \frac{2 π}{3} + 2 πn : x = \frac{4 π}{3} + 4 πn

\frac{1}{2} x = \frac{2 π}{3} + 2 πn

Multiply both sides by

2

\frac{1}{2} x = \frac{2 π}{3} + 2 πn

Multiply both sides by

2

2 \cdot \frac{1}{2} x = 2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x = 2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x : x

2 \cdot \frac{1}{2} x

Multiply fractions:

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

= \frac{1 \cdot 2}{2} x

Cancel the common factor:

2

= x \cdot 1

Multiply:

x \cdot 1 = x

= x

Simplify

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

2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn

2 \cdot \frac{2 π}{3} = \frac{4 π}{3}

2 \cdot \frac{2 π}{3}

Multiply fractions:

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

= \frac{2 π 2}{3}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{4 π}{3} + 4 πn

x = \frac{4 π}{3} + 4 πn

x = \frac{4 π}{3} + 4 πn

x = \frac{4 π}{3} + 4 πn

Solve

\frac{1}{2} x = \frac{4 π}{3} + 2 πn : x = \frac{8 π}{3} + 4 πn

\frac{1}{2} x = \frac{4 π}{3} + 2 πn

Multiply both sides by

2

\frac{1}{2} x = \frac{4 π}{3} + 2 πn

Multiply both sides by

2

2 \cdot \frac{1}{2} x = 2 \cdot \frac{4 π}{3} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x = 2 \cdot \frac{4 π}{3} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x : x

2 \cdot \frac{1}{2} x

Multiply fractions:

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

= \frac{1 \cdot 2}{2} x

Cancel the common factor:

2

= x \cdot 1

Multiply:

x \cdot 1 = x

= x

Simplify

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

2 \cdot \frac{4 π}{3} + 2 \cdot 2 πn

2 \cdot \frac{4 π}{3} = \frac{8 π}{3}

2 \cdot \frac{4 π}{3}

Multiply fractions:

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

= \frac{4 π 2}{3}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{8 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{8 π}{3} + 4 πn

x = \frac{8 π}{3} + 4 πn

x = \frac{8 π}{3} + 4 πn

x = \frac{8 π}{3} + 4 πn

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

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2tan(θ)+4=tan(θ)+5

2 tan (θ) + 4 = tan (θ) + 5

solvefor x,sin(x)= 4/5 solve for

x, sin (x) = \frac{4}{5}

tan(x)=(-25}{\frac{-80-50)/2}

tan (x) = \frac{- 25}{\frac{- 80 - 50}{2}}

Frequently Asked Questions (FAQ)

What is the general solution for 8cos(1/2 x)+4=0 ?
The general solution for 8cos(1/2 x)+4=0 is x=(4pi)/3+4pin,x=(8pi)/3+4pin