What is the general solution for cos(0.5x)=0.5,-pi<= x<= pi ?

Q: What is the general solution for cos(0.5x)=0.5,-pi<= x<= pi ?

The general solution for cos(0.5x)=0.5,-pi<= x<= pi is x=(2pi)/3 ,x=-(2pi)/3

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cos(0.5x)=0.5,-pi<= x<= pi

Solution

cos (0.5 x) = 0.5, - π \leq x \leq π

Solution

x = \frac{2 π}{3}, x = - \frac{2 π}{3}

Degrees

x = 12 0^{\circ}, x = - 12 0^{\circ}

Solution steps

cos (0.5 x) = 0.5, - π \leq x \leq π

General solutions for

cos (0.5 x) = 0.5

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}

0.5 x = \frac{π}{3} + 2 πn, 0.5 x = \frac{5 π}{3} + 2 πn

0.5 x = \frac{π}{3} + 2 πn, 0.5 x = \frac{5 π}{3} + 2 πn

Solve

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

0.5 x = \frac{π}{3} + 2 πn

Multiply both sides by

10

0.5 x = \frac{π}{3} + 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

0.5 x \cdot 10 = \frac{π}{3} \cdot 10 + 2 πn \cdot 10

Refine

5 x = \frac{10 π}{3} + 20 πn

5 x = \frac{10 π}{3} + 20 πn

Divide both sides by

5

5 x = \frac{10 π}{3} + 20 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{\frac{10 π}{3}}{5} + \frac{20 πn}{5}

Simplify

\frac{5 x}{5} = \frac{\frac{10 π}{3}}{5} + \frac{20 πn}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{\frac{10 π}{3}}{5} + \frac{20 πn}{5} : \frac{2 π}{3} + 4 πn

\frac{\frac{10 π}{3}}{5} + \frac{20 πn}{5}

\frac{\frac{10 π}{3}}{5} = \frac{2 π}{3}

\frac{\frac{10 π}{3}}{5}

Apply the fraction rule:

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

= \frac{10 π}{3 \cdot 5}

Multiply the numbers:

3 \cdot 5 = 15

= \frac{10 π}{15}

Cancel the common factor:

5

= \frac{2 π}{3}

\frac{20 πn}{5} = 4 πn

\frac{20 πn}{5}

Divide the numbers:

\frac{20}{5} = 4

= 4 πn

= \frac{2 π}{3} + 4 πn

x = \frac{2 π}{3} + 4 πn

x = \frac{2 π}{3} + 4 πn

x = \frac{2 π}{3} + 4 πn

Solve

0.5 x = \frac{5 π}{3} + 2 πn : x = \frac{10 π}{3} + 4 πn

0.5 x = \frac{5 π}{3} + 2 πn

Multiply both sides by

10

0.5 x = \frac{5 π}{3} + 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

0.5 x \cdot 10 = \frac{5 π}{3} \cdot 10 + 2 πn \cdot 10

Refine

5 x = \frac{50 π}{3} + 20 πn

5 x = \frac{50 π}{3} + 20 πn

Divide both sides by

5

5 x = \frac{50 π}{3} + 20 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{\frac{50 π}{3}}{5} + \frac{20 πn}{5}

Simplify

\frac{5 x}{5} = \frac{\frac{50 π}{3}}{5} + \frac{20 πn}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{\frac{50 π}{3}}{5} + \frac{20 πn}{5} : \frac{10 π}{3} + 4 πn

\frac{\frac{50 π}{3}}{5} + \frac{20 πn}{5}

\frac{\frac{50 π}{3}}{5} = \frac{10 π}{3}

\frac{\frac{50 π}{3}}{5}

Apply the fraction rule:

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

= \frac{50 π}{3 \cdot 5}

Multiply the numbers:

3 \cdot 5 = 15

= \frac{50 π}{15}

Cancel the common factor:

5

= \frac{10 π}{3}

\frac{20 πn}{5} = 4 πn

\frac{20 πn}{5}

Divide the numbers:

\frac{20}{5} = 4

= 4 πn

= \frac{10 π}{3} + 4 πn

x = \frac{10 π}{3} + 4 πn

x = \frac{10 π}{3} + 4 πn

x = \frac{10 π}{3} + 4 πn

x = \frac{2 π}{3} + 4 πn, x = \frac{10 π}{3} + 4 πn

Solutions for the range

- π \leq x \leq π

x = \frac{2 π}{3}, x = - \frac{2 π}{3}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(0.5x)=0.5,-pi<= x<= pi ?
The general solution for cos(0.5x)=0.5,-pi<= x<= pi is x=(2pi)/3 ,x=-(2pi)/3