What is the general solution for cos(5x+3)= 1/2 ?

The general solution for cos(5x+3)= 1/2 is x=-3/5+pi/(15)+(2pin)/5 ,x=-3/5+pi/3+(2pin)/5

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cos(5x+3)= 1/2

Solution

cos (5 x + 3) = \frac{1}{2}

Solution

x = - \frac{3}{5} + \frac{π}{15} + \frac{2 πn}{5}, x = - \frac{3}{5} + \frac{π}{3} + \frac{2 πn}{5}

Degrees

x = - 22.37746 \dots^{\circ} + 7 2^{\circ} n, x = 25.62253 \dots^{\circ} + 7 2^{\circ} n

Solution steps

cos (5 x + 3) = \frac{1}{2}

General solutions for

cos (5 x + 3) = \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}

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

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

Solve

5 x + 3 = \frac{π}{3} + 2 πn : x = - \frac{3}{5} + \frac{π}{15} + \frac{2 πn}{5}

5 x + 3 = \frac{π}{3} + 2 πn

Move

3

to the right side

5 x + 3 = \frac{π}{3} + 2 πn

Subtract

3

from both sides

5 x + 3 - 3 = \frac{π}{3} + 2 πn - 3

Simplify

5 x = \frac{π}{3} + 2 πn - 3

5 x = \frac{π}{3} + 2 πn - 3

Divide both sides by

5

5 x = \frac{π}{3} + 2 πn - 3

Divide both sides by

5

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

Simplify

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

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{\frac{π}{3}}{5} + \frac{2 πn}{5} - \frac{3}{5} : - \frac{3}{5} + \frac{π}{15} + \frac{2 πn}{5}

\frac{\frac{π}{3}}{5} + \frac{2 πn}{5} - \frac{3}{5}

Group like terms

= - \frac{3}{5} + \frac{2 πn}{5} + \frac{\frac{π}{3}}{5}

\frac{\frac{π}{3}}{5} = \frac{π}{15}

\frac{\frac{π}{3}}{5}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 5}

Multiply the numbers:

3 \cdot 5 = 15

= \frac{π}{15}

= - \frac{3}{5} + \frac{2 πn}{5} + \frac{π}{15}

Group like terms

= - \frac{3}{5} + \frac{π}{15} + \frac{2 πn}{5}

x = - \frac{3}{5} + \frac{π}{15} + \frac{2 πn}{5}

x = - \frac{3}{5} + \frac{π}{15} + \frac{2 πn}{5}

x = - \frac{3}{5} + \frac{π}{15} + \frac{2 πn}{5}

Solve

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

5 x + 3 = \frac{5 π}{3} + 2 πn

Move

3

to the right side

5 x + 3 = \frac{5 π}{3} + 2 πn

Subtract

3

from both sides

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

Simplify

5 x = \frac{5 π}{3} + 2 πn - 3

5 x = \frac{5 π}{3} + 2 πn - 3

Divide both sides by

5

5 x = \frac{5 π}{3} + 2 πn - 3

Divide both sides by

5

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

Simplify

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

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{\frac{5 π}{3}}{5} + \frac{2 πn}{5} - \frac{3}{5} : - \frac{3}{5} + \frac{π}{3} + \frac{2 πn}{5}

\frac{\frac{5 π}{3}}{5} + \frac{2 πn}{5} - \frac{3}{5}

Group like terms

= - \frac{3}{5} + \frac{2 πn}{5} + \frac{\frac{5 π}{3}}{5}

\frac{\frac{5 π}{3}}{5} = \frac{π}{3}

\frac{\frac{5 π}{3}}{5}

Apply the fraction rule:

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

= \frac{5 π}{3 \cdot 5}

Multiply the numbers:

3 \cdot 5 = 15

= \frac{5 π}{15}

Cancel the common factor:

5

= \frac{π}{3}

= - \frac{3}{5} + \frac{2 πn}{5} + \frac{π}{3}

Group like terms

= - \frac{3}{5} + \frac{π}{3} + \frac{2 πn}{5}

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

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

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

x = - \frac{3}{5} + \frac{π}{15} + \frac{2 πn}{5}, x = - \frac{3}{5} + \frac{π}{3} + \frac{2 πn}{5}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(5x+3)= 1/2 ?
The general solution for cos(5x+3)= 1/2 is x=-3/5+pi/(15)+(2pin)/5 ,x=-3/5+pi/3+(2pin)/5