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Popular Trigonometry >

solvefor k,cos(kl)=0

Solution

solvefor

Solution

k = \frac{π}{2 l} + \frac{2 πn}{l}, k = \frac{3 π}{2 l} + \frac{2 πn}{l}; l \neq = 0

Solution steps

cos (k l) = 0

General solutions for

cos (k l) = 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}

k l = \frac{π}{2} + 2 πn, k l = \frac{3 π}{2} + 2 πn

k l = \frac{π}{2} + 2 πn, k l = \frac{3 π}{2} + 2 πn

Solve

k l = \frac{π}{2} + 2 πn : k = \frac{π}{2 l} + \frac{2 πn}{l}; l \neq = 0

k l = \frac{π}{2} + 2 πn

Divide both sides by

l; l \neq = 0

k l = \frac{π}{2} + 2 πn

Divide both sides by

l; l \neq = 0

\frac{k l}{l} = \frac{\frac{π}{2}}{l} + \frac{2 πn}{l}; l \neq = 0

Simplify

\frac{k l}{l} = \frac{\frac{π}{2}}{l} + \frac{2 πn}{l}

Simplify

\frac{k l}{l} : k

\frac{k l}{l}

Cancel the common factor:

l

= k

Simplify

\frac{\frac{π}{2}}{l} + \frac{2 πn}{l} : \frac{π}{2 l} + \frac{2 πn}{l}

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

Apply the fraction rule:

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

= \frac{π}{2 l} + \frac{2 πn}{l}

k = \frac{π}{2 l} + \frac{2 πn}{l}; l \neq = 0

k = \frac{π}{2 l} + \frac{2 πn}{l}; l \neq = 0

k = \frac{π}{2 l} + \frac{2 πn}{l}; l \neq = 0

Solve

k l = \frac{3 π}{2} + 2 πn : k = \frac{3 π}{2 l} + \frac{2 πn}{l}; l \neq = 0

k l = \frac{3 π}{2} + 2 πn

Divide both sides by

l; l \neq = 0

k l = \frac{3 π}{2} + 2 πn

Divide both sides by

l; l \neq = 0

\frac{k l}{l} = \frac{\frac{3 π}{2}}{l} + \frac{2 πn}{l}; l \neq = 0

Simplify

\frac{k l}{l} = \frac{\frac{3 π}{2}}{l} + \frac{2 πn}{l}

Simplify

\frac{k l}{l} : k

\frac{k l}{l}

Cancel the common factor:

l

= k

Simplify

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

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

Apply the fraction rule:

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

= \frac{3 π}{2 l} + \frac{2 πn}{l}

k = \frac{3 π}{2 l} + \frac{2 πn}{l}; l \neq = 0

k = \frac{3 π}{2 l} + \frac{2 πn}{l}; l \neq = 0

k = \frac{3 π}{2 l} + \frac{2 πn}{l}; l \neq = 0

k = \frac{π}{2 l} + \frac{2 πn}{l}, k = \frac{3 π}{2 l} + \frac{2 πn}{l}; l \neq = 0

Graph

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