What is the general solution for 3sin(θ/4)-3/2 =0,0<= θ<= 2pi ?

Q: What is the general solution for 3sin(θ/4)-3/2 =0,0<= θ<= 2pi ?

The general solution for 3sin(θ/4)-3/2 =0,0<= θ<= 2pi is θ=(2pi)/3

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3sin(θ/4)-3/2 =0,0<= θ<= 2pi

Solution

3 sin (\frac{θ}{4}) - \frac{3}{2} = 0, 0 \leq θ \leq 2 π

Solution

θ = \frac{2 π}{3}

Degrees

θ = 12 0^{\circ}

Solution steps

3 sin (\frac{θ}{4}) - \frac{3}{2} = 0, 0 \leq θ \leq 2 π

Move

\frac{3}{2}

to the right side

3 sin (\frac{θ}{4}) - \frac{3}{2} = 0

Add

\frac{3}{2}

to both sides

3 sin (\frac{θ}{4}) - \frac{3}{2} + \frac{3}{2} = 0 + \frac{3}{2}

Simplify

3 sin (\frac{θ}{4}) = \frac{3}{2}

3 sin (\frac{θ}{4}) = \frac{3}{2}

Divide both sides by

3

3 sin (\frac{θ}{4}) = \frac{3}{2}

Divide both sides by

3

\frac{3 sin ( \frac{θ}{4} )}{3} = \frac{\frac{3}{2}}{3}

Simplify

\frac{3 sin ( \frac{θ}{4} )}{3} = \frac{\frac{3}{2}}{3}

Simplify

\frac{3 sin ( \frac{θ}{4} )}{3} : sin (\frac{θ}{4})

\frac{3 sin ( \frac{θ}{4} )}{3}

Divide the numbers:

\frac{3}{3} = 1

= sin (\frac{θ}{4})

Simplify

\frac{\frac{3}{2}}{3} : \frac{1}{2}

\frac{\frac{3}{2}}{3}

Apply the fraction rule:

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

= \frac{3}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{3}{6}

Cancel the common factor:

3

= \frac{1}{2}

sin (\frac{θ}{4}) = \frac{1}{2}

sin (\frac{θ}{4}) = \frac{1}{2}

sin (\frac{θ}{4}) = \frac{1}{2}

General solutions for

sin (\frac{θ}{4}) = \frac{1}{2}

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}

\frac{θ}{4} = \frac{π}{6} + 2 πn, \frac{θ}{4} = \frac{5 π}{6} + 2 πn

\frac{θ}{4} = \frac{π}{6} + 2 πn, \frac{θ}{4} = \frac{5 π}{6} + 2 πn

Solve

\frac{θ}{4} = \frac{π}{6} + 2 πn : θ = \frac{2 π}{3} + 8 πn

\frac{θ}{4} = \frac{π}{6} + 2 πn

Multiply both sides by

4

\frac{θ}{4} = \frac{π}{6} + 2 πn

Multiply both sides by

4

\frac{4 θ}{4} = 4 \cdot \frac{π}{6} + 4 \cdot 2 πn

Simplify

\frac{4 θ}{4} = 4 \cdot \frac{π}{6} + 4 \cdot 2 πn

Simplify

\frac{4 θ}{4} : θ

\frac{4 θ}{4}

Divide the numbers:

\frac{4}{4} = 1

= θ

Simplify

4 \cdot \frac{π}{6} + 4 \cdot 2 πn : \frac{2 π}{3} + 8 πn

4 \cdot \frac{π}{6} + 4 \cdot 2 πn

4 \cdot \frac{π}{6} = \frac{2 π}{3}

4 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 4}{6}

Cancel the common factor:

2

= \frac{2 π}{3}

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= \frac{2 π}{3} + 8 πn

θ = \frac{2 π}{3} + 8 πn

θ = \frac{2 π}{3} + 8 πn

θ = \frac{2 π}{3} + 8 πn

Solve

\frac{θ}{4} = \frac{5 π}{6} + 2 πn : θ = \frac{10 π}{3} + 8 πn

\frac{θ}{4} = \frac{5 π}{6} + 2 πn

Multiply both sides by

4

\frac{θ}{4} = \frac{5 π}{6} + 2 πn

Multiply both sides by

4

\frac{4 θ}{4} = 4 \cdot \frac{5 π}{6} + 4 \cdot 2 πn

Simplify

\frac{4 θ}{4} = 4 \cdot \frac{5 π}{6} + 4 \cdot 2 πn

Simplify

\frac{4 θ}{4} : θ

\frac{4 θ}{4}

Divide the numbers:

\frac{4}{4} = 1

= θ

Simplify

4 \cdot \frac{5 π}{6} + 4 \cdot 2 πn : \frac{10 π}{3} + 8 πn

4 \cdot \frac{5 π}{6} + 4 \cdot 2 πn

4 \cdot \frac{5 π}{6} = \frac{10 π}{3}

4 \cdot \frac{5 π}{6}

Multiply fractions:

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

= \frac{5 π 4}{6}

Multiply the numbers:

5 \cdot 4 = 20

= \frac{20 π}{6}

Cancel the common factor:

2

= \frac{10 π}{3}

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= \frac{10 π}{3} + 8 πn

θ = \frac{10 π}{3} + 8 πn

θ = \frac{10 π}{3} + 8 πn

θ = \frac{10 π}{3} + 8 πn

θ = \frac{2 π}{3} + 8 πn, θ = \frac{10 π}{3} + 8 πn

Solutions for the range

0 \leq θ \leq 2 π

θ = \frac{2 π}{3}

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Frequently Asked Questions (FAQ)

What is the general solution for 3sin(θ/4)-3/2 =0,0<= θ<= 2pi ?
The general solution for 3sin(θ/4)-3/2 =0,0<= θ<= 2pi is θ=(2pi)/3