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

The general solution for cos(4x)= 1/2 is x= pi/(12)+(pin)/2 ,x=(5pi}{12}+\frac{pin)/2

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cos(4x)= 1/2

Solution

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

Solution

x = \frac{π}{12} + \frac{πn}{2}, x = \frac{5 π}{12} + \frac{πn}{2}

Degrees

x = 1 5^{\circ} + 9 0^{\circ} n, x = 7 5^{\circ} + 9 0^{\circ} n

Solution steps

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

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{π}{3} + 2 πn, 4 x = \frac{5 π}{3} + 2 πn

4 x = \frac{π}{3} + 2 πn, 4 x = \frac{5 π}{3} + 2 πn

Solve

4 x = \frac{π}{3} + 2 πn : x = \frac{π}{12} + \frac{πn}{2}

4 x = \frac{π}{3} + 2 πn

Divide both sides by

4

4 x = \frac{π}{3} + 2 πn

Divide both sides by

4

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

Simplify

\frac{4 x}{4} = \frac{\frac{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{π}{3}}{4} + \frac{2 πn}{4} : \frac{π}{12} + \frac{πn}{2}

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

\frac{\frac{π}{3}}{4} = \frac{π}{12}

\frac{\frac{π}{3}}{4}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 4}

Multiply the numbers:

3 \cdot 4 = 12

= \frac{π}{12}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{π}{12} + \frac{πn}{2}

x = \frac{π}{12} + \frac{πn}{2}

x = \frac{π}{12} + \frac{πn}{2}

x = \frac{π}{12} + \frac{πn}{2}

Solve

4 x = \frac{5 π}{3} + 2 πn : x = \frac{5 π}{12} + \frac{πn}{2}

4 x = \frac{5 π}{3} + 2 πn

Divide both sides by

4

4 x = \frac{5 π}{3} + 2 πn

Divide both sides by

4

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

Simplify

\frac{4 x}{4} = \frac{\frac{5 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{5 π}{3}}{4} + \frac{2 πn}{4} : \frac{5 π}{12} + \frac{πn}{2}

\frac{\frac{5 π}{3}}{4} + \frac{2 πn}{4}

\frac{\frac{5 π}{3}}{4} = \frac{5 π}{12}

\frac{\frac{5 π}{3}}{4}

Apply the fraction rule:

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

= \frac{5 π}{3 \cdot 4}

Multiply the numbers:

3 \cdot 4 = 12

= \frac{5 π}{12}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{5 π}{12} + \frac{πn}{2}

x = \frac{5 π}{12} + \frac{πn}{2}

x = \frac{5 π}{12} + \frac{πn}{2}

x = \frac{5 π}{12} + \frac{πn}{2}

x = \frac{π}{12} + \frac{πn}{2}, x = \frac{5 π}{12} + \frac{πn}{2}

Popular Examples

2cos^2(x)-5cos(x)+2=0

2 cos^{2} (x) - 5 cos (x) + 2 = 0

sec(x)=-sqrt(2)

sec (x) = - 2

2sin^2(θ)-cos(θ)=1

2 sin^{2} (θ) - cos (θ) = 1

3sin^2(x)-7sin(x)+2=0

3 sin^{2} (x) - 7 sin (x) + 2 = 0

tan(θ)= 4/3

tan (θ) = \frac{4}{3}

Frequently Asked Questions (FAQ)

What is the general solution for cos(4x)= 1/2 ?
The general solution for cos(4x)= 1/2 is x= pi/(12)+(pin)/2 ,x=(5pi}{12}+\frac{pin)/2