What is the general solution for (sin((2*5+1)*w/2))/(sin(w/2))=0 ?

The general solution for (sin((2*5+1)*w/2))/(sin(w/2))=0 is w=(4pin)/(11),w=(2pi)/(11)+(4pin)/(11)

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(sin((25+1)w/2))/(sin(w/2))=0

Solution

\frac{sin ( ( 2 \cdot 5 + 1 ) \cdot \frac{w}{2} )}{sin ( \frac{w}{2} )} = 0

Solution

w = \frac{4 πn}{11}, w = \frac{2 π}{11} + \frac{4 πn}{11}

Degrees

w = 0^{\circ} + 65.45454 \dots^{\circ} n, w = 32.72727 \dots^{\circ} + 65.45454 \dots^{\circ} n

Solution steps

\frac{sin ( ( 2 \cdot 5 + 1 ) \frac{w}{2} )}{sin ( \frac{w}{2} )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

sin ((2 \cdot 5 + 1) \frac{w}{2}) = 0

General solutions for

sin ((25 + 1) \frac{w}{2}) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

(25 + 1) \frac{w}{2} = 0 + 2 πn, (25 + 1) \frac{w}{2} = π + 2 πn

(25 + 1) \frac{w}{2} = 0 + 2 πn, (25 + 1) \frac{w}{2} = π + 2 πn

Solve

(25 + 1) \frac{w}{2} = 0 + 2 πn : w = \frac{4 πn}{11}

(2 \cdot 5 + 1) \frac{w}{2} = 0 + 2 πn

Refine

(2 \cdot 5 + 1) \frac{w}{2} : \frac{11 w}{2}

(2 \cdot 5 + 1) \frac{w}{2}

Multiply the numbers:

2 \cdot 5 = 10

= (10 + 1) \frac{w}{2}

Add the numbers:

10 + 1 = 11

= 11 \cdot \frac{w}{2}

Multiply fractions:

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

= \frac{w \cdot 11}{2}

Expand

0 + 2 πn : 2 πn

0 + 2 πn

0 + 2 πn = 2 πn

= 2 πn

\frac{11 w}{2} = 2 πn

Multiply both sides by

2

\frac{11 w}{2} = 2 πn

Multiply both sides by

2

\frac{11 w}{2} \cdot 2 = 2 πn \cdot 2

Simplify

\frac{11 w}{2} \cdot 2 = 2 πn \cdot 2

Simplify

\frac{11 w}{2} \cdot 2 : 11 w

\frac{11 w}{2} \cdot 2

Multiply fractions:

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

= \frac{11 w \cdot 2}{2}

Cancel the common factor:

2

= 11 w

Simplify

2 πn \cdot 2 : 4 πn

2 πn \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

11 w = 4 πn

11 w = 4 πn

11 w = 4 πn

Divide both sides by

11

11 w = 4 πn

Divide both sides by

11

\frac{11 w}{11} = \frac{4 πn}{11}

Simplify

w = \frac{4 πn}{11}

w = \frac{4 πn}{11}

Solve

(25 + 1) \frac{w}{2} = π + 2 πn : w = \frac{2 π}{11} + \frac{4 πn}{11}

(2 \cdot 5 + 1) \frac{w}{2} = π + 2 πn

Refine

(2 \cdot 5 + 1) \frac{w}{2} : \frac{11 w}{2}

(2 \cdot 5 + 1) \frac{w}{2}

Multiply the numbers:

2 \cdot 5 = 10

= (10 + 1) \frac{w}{2}

Add the numbers:

10 + 1 = 11

= 11 \cdot \frac{w}{2}

Multiply fractions:

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

= \frac{w \cdot 11}{2}

\frac{11 w}{2} = π + 2 πn

Multiply both sides by

2

\frac{11 w}{2} = π + 2 πn

Multiply both sides by

2

\frac{11 w}{2} \cdot 2 = π 2 + 2 πn \cdot 2

Simplify

\frac{11 w}{2} \cdot 2 = π 2 + 2 πn \cdot 2

Simplify

\frac{11 w}{2} \cdot 2 : 11 w

\frac{11 w}{2} \cdot 2

Multiply fractions:

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

= \frac{11 w \cdot 2}{2}

Cancel the common factor:

2

= 11 w

Simplify

π 2 : 2 π

π 2

Apply the commutative law:

π 2 = 2 π

2 π

Simplify

2 πn \cdot 2 : 4 πn

2 πn \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

11 w = 2 π + 4 πn

11 w = 2 π + 4 πn

11 w = 2 π + 4 πn

Divide both sides by

11

11 w = 2 π + 4 πn

Divide both sides by

11

\frac{11 w}{11} = \frac{2 π}{11} + \frac{4 πn}{11}

Simplify

w = \frac{2 π}{11} + \frac{4 πn}{11}

w = \frac{2 π}{11} + \frac{4 πn}{11}

w = \frac{4 πn}{11}, w = \frac{2 π}{11} + \frac{4 πn}{11}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for (sin((2*5+1)*w/2))/(sin(w/2))=0 ?
The general solution for (sin((2*5+1)*w/2))/(sin(w/2))=0 is w=(4pin)/(11),w=(2pi)/(11)+(4pin)/(11)