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Popular >

cos(4x)=-1/2 ,0<= x<= 2pi

Solution

cos (4 x) = - \frac{1}{2}, 0 \leq x \leq 2 π

Solution

x = \frac{π}{6}, x = \frac{π}{3}, x = \frac{2 π}{3}, x = \frac{5 π}{6}, x = \frac{7 π}{6}, x = \frac{4 π}{3}, x = \frac{5 π}{3}, x = \frac{11 π}{6}

Degrees

x = 3 0^{\circ}, x = 6 0^{\circ}, x = 12 0^{\circ}, x = 15 0^{\circ}, x = 21 0^{\circ}, x = 24 0^{\circ}, x = 30 0^{\circ}, x = 33 0^{\circ}

Solution steps

cos (4 x) = - \frac{1}{2}, 0 \leq x \leq 2 π

General solutions for

cos (4 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}

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

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

Solve

4 x = \frac{2 π}{3} + 2 πn : x = \frac{π}{6} + \frac{πn}{2}

4 x = \frac{2 π}{3} + 2 πn

Divide both sides by

4

4 x = \frac{2 π}{3} + 2 πn

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{\frac{2 π}{3}}{4} + \frac{2 πn}{4} : \frac{π}{6} + \frac{πn}{2}

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

\frac{\frac{2 π}{3}}{4} = \frac{π}{6}

\frac{\frac{2 π}{3}}{4}

Apply the fraction rule:

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

= \frac{2 π}{3 \cdot 4}

Multiply the numbers:

3 \cdot 4 = 12

= \frac{2 π}{12}

Cancel the common factor:

2

= \frac{π}{6}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{π}{6} + \frac{πn}{2}

x = \frac{π}{6} + \frac{πn}{2}

x = \frac{π}{6} + \frac{πn}{2}

x = \frac{π}{6} + \frac{πn}{2}

Solve

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

4 x = \frac{4 π}{3} + 2 πn

Divide both sides by

4

4 x = \frac{4 π}{3} + 2 πn

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

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

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

\frac{\frac{4 π}{3}}{4} = \frac{π}{3}

\frac{\frac{4 π}{3}}{4}

Apply the fraction rule:

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

= \frac{4 π}{3 \cdot 4}

Multiply the numbers:

3 \cdot 4 = 12

= \frac{4 π}{12}

Cancel the common factor:

4

= \frac{π}{3}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{π}{3} + \frac{πn}{2}

x = \frac{π}{3} + \frac{πn}{2}

x = \frac{π}{3} + \frac{πn}{2}

x = \frac{π}{3} + \frac{πn}{2}

x = \frac{π}{6} + \frac{πn}{2}, x = \frac{π}{3} + \frac{πn}{2}

Solutions for the range

0 \leq x \leq 2 π

x = \frac{π}{6}, x = \frac{π}{3}, x = \frac{2 π}{3}, x = \frac{5 π}{6}, x = \frac{7 π}{6}, x = \frac{4 π}{3}, x = \frac{5 π}{3}, x = \frac{11 π}{6}

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