What is the general solution for cos(4w)=0 ?

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

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cos(4w)=0

Solution

cos (4 w) = 0

Solution

w = \frac{π}{8} + \frac{πn}{2}, w = \frac{3 π}{8} + \frac{πn}{2}

Degrees

w = 22. 5^{\circ} + 9 0^{\circ} n, w = 67. 5^{\circ} + 9 0^{\circ} n

Solution steps

cos (4 w) = 0

General solutions for

cos (4 w) = 0

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

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

Solve

4 w = \frac{π}{2} + 2 πn : w = \frac{π}{8} + \frac{πn}{2}

4 w = \frac{π}{2} + 2 πn

Divide both sides by

4

4 w = \frac{π}{2} + 2 πn

Divide both sides by

4

\frac{4 w}{4} = \frac{\frac{π}{2}}{4} + \frac{2 πn}{4}

Simplify

\frac{4 w}{4} = \frac{\frac{π}{2}}{4} + \frac{2 πn}{4}

Simplify

\frac{4 w}{4} : w

\frac{4 w}{4}

Divide the numbers:

\frac{4}{4} = 1

= w

Simplify

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

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

\frac{\frac{π}{2}}{4} = \frac{π}{8}

\frac{\frac{π}{2}}{4}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{π}{8}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{π}{8} + \frac{πn}{2}

w = \frac{π}{8} + \frac{πn}{2}

w = \frac{π}{8} + \frac{πn}{2}

w = \frac{π}{8} + \frac{πn}{2}

Solve

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

4 w = \frac{3 π}{2} + 2 πn

Divide both sides by

4

4 w = \frac{3 π}{2} + 2 πn

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 w}{4} : w

\frac{4 w}{4}

Divide the numbers:

\frac{4}{4} = 1

= w

Simplify

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

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

\frac{\frac{3 π}{2}}{4} = \frac{3 π}{8}

\frac{\frac{3 π}{2}}{4}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{3 π}{8}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{3 π}{8} + \frac{πn}{2}

w = \frac{3 π}{8} + \frac{πn}{2}

w = \frac{3 π}{8} + \frac{πn}{2}

w = \frac{3 π}{8} + \frac{πn}{2}

w = \frac{π}{8} + \frac{πn}{2}, w = \frac{3 π}{8} + \frac{πn}{2}

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cos(θ)= 2/11

cos (θ) = \frac{2}{11}

Frequently Asked Questions (FAQ)

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