What is the general solution for cos(15θ)=0 ?

The general solution for cos(15θ)=0 is θ= pi/(30)+(2pin)/(15),θ= pi/(10)+(2pin)/(15)

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cos(15θ)=0

Solution

cos (15 θ) = 0

Solution

θ = \frac{π}{30} + \frac{2 πn}{15}, θ = \frac{π}{10} + \frac{2 πn}{15}

Degrees

θ = 6^{\circ} + 2 4^{\circ} n, θ = 1 8^{\circ} + 2 4^{\circ} n

Solution steps

cos (15 θ) = 0

General solutions for

cos (15 θ) = 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}

15 θ = \frac{π}{2} + 2 πn, 15 θ = \frac{3 π}{2} + 2 πn

15 θ = \frac{π}{2} + 2 πn, 15 θ = \frac{3 π}{2} + 2 πn

Solve

15 θ = \frac{π}{2} + 2 πn : θ = \frac{π}{30} + \frac{2 πn}{15}

15 θ = \frac{π}{2} + 2 πn

Divide both sides by

15

15 θ = \frac{π}{2} + 2 πn

Divide both sides by

15

\frac{15 θ}{15} = \frac{\frac{π}{2}}{15} + \frac{2 πn}{15}

Simplify

\frac{15 θ}{15} = \frac{\frac{π}{2}}{15} + \frac{2 πn}{15}

Simplify

\frac{15 θ}{15} : θ

\frac{15 θ}{15}

Divide the numbers:

\frac{15}{15} = 1

= θ

Simplify

\frac{\frac{π}{2}}{15} + \frac{2 πn}{15} : \frac{π}{30} + \frac{2 πn}{15}

\frac{\frac{π}{2}}{15} + \frac{2 πn}{15}

\frac{\frac{π}{2}}{15} = \frac{π}{30}

\frac{\frac{π}{2}}{15}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 15}

Multiply the numbers:

2 \cdot 15 = 30

= \frac{π}{30}

= \frac{π}{30} + \frac{2 πn}{15}

θ = \frac{π}{30} + \frac{2 πn}{15}

θ = \frac{π}{30} + \frac{2 πn}{15}

θ = \frac{π}{30} + \frac{2 πn}{15}

Solve

15 θ = \frac{3 π}{2} + 2 πn : θ = \frac{π}{10} + \frac{2 πn}{15}

15 θ = \frac{3 π}{2} + 2 πn

Divide both sides by

15

15 θ = \frac{3 π}{2} + 2 πn

Divide both sides by

15

\frac{15 θ}{15} = \frac{\frac{3 π}{2}}{15} + \frac{2 πn}{15}

Simplify

\frac{15 θ}{15} = \frac{\frac{3 π}{2}}{15} + \frac{2 πn}{15}

Simplify

\frac{15 θ}{15} : θ

\frac{15 θ}{15}

Divide the numbers:

\frac{15}{15} = 1

= θ

Simplify

\frac{\frac{3 π}{2}}{15} + \frac{2 πn}{15} : \frac{π}{10} + \frac{2 πn}{15}

\frac{\frac{3 π}{2}}{15} + \frac{2 πn}{15}

\frac{\frac{3 π}{2}}{15} = \frac{π}{10}

\frac{\frac{3 π}{2}}{15}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 15}

Multiply the numbers:

2 \cdot 15 = 30

= \frac{3 π}{30}

Cancel the common factor:

3

= \frac{π}{10}

= \frac{π}{10} + \frac{2 πn}{15}

θ = \frac{π}{10} + \frac{2 πn}{15}

θ = \frac{π}{10} + \frac{2 πn}{15}

θ = \frac{π}{10} + \frac{2 πn}{15}

θ = \frac{π}{30} + \frac{2 πn}{15}, θ = \frac{π}{10} + \frac{2 πn}{15}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos(15θ)=0 ?
The general solution for cos(15θ)=0 is θ= pi/(30)+(2pin)/(15),θ= pi/(10)+(2pin)/(15)