What is the general solution for sin((3x)/2+45)=0 ?

The general solution for sin((3x)/2+45)=0 is x=(720n)/3-30,x=-30+120+(720n)/3

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sin((3x)/2+45)=0

Solution

sin (\frac{3 x}{2} + 4 5^{\circ}) = 0

Solution

x = \frac{72 0 ^{\circ} n}{3} - 3 0^{\circ}, x = - 3 0^{\circ} + 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

Radians

x = - \frac{π}{6} + \frac{4 π}{3} n, x = - \frac{π}{6} + \frac{2 π}{3} + \frac{4 π}{3} n

Solution steps

sin (\frac{3 x}{2} + 4 5^{\circ}) = 0

General solutions for

sin (\frac{3 x}{2} + 4 5^{\circ}) = 0

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{3 x}{2} + 4 5^{\circ} = 0 + 36 0^{\circ} n, \frac{3 x}{2} + 4 5^{\circ} = 18 0^{\circ} + 36 0^{\circ} n

\frac{3 x}{2} + 4 5^{\circ} = 0 + 36 0^{\circ} n, \frac{3 x}{2} + 4 5^{\circ} = 18 0^{\circ} + 36 0^{\circ} n

Solve

\frac{3 x}{2} + 4 5^{\circ} = 0 + 36 0^{\circ} n : x = \frac{72 0 ^{\circ} n}{3} - 3 0^{\circ}

\frac{3 x}{2} + 4 5^{\circ} = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

\frac{3 x}{2} + 4 5^{\circ} = 36 0^{\circ} n

Move

4 5^{\circ}

to the right side

\frac{3 x}{2} + 4 5^{\circ} = 36 0^{\circ} n

Subtract

4 5^{\circ}

from both sides

\frac{3 x}{2} + 4 5^{\circ} - 4 5^{\circ} = 36 0^{\circ} n - 4 5^{\circ}

Simplify

\frac{3 x}{2} = 36 0^{\circ} n - 4 5^{\circ}

\frac{3 x}{2} = 36 0^{\circ} n - 4 5^{\circ}

Multiply both sides by

2

\frac{3 x}{2} = 36 0^{\circ} n - 4 5^{\circ}

Multiply both sides by

2

\frac{2 \cdot 3 x}{2} = 2 \cdot 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

Simplify

\frac{2 \cdot 3 x}{2} = 2 \cdot 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

Simplify

\frac{2 \cdot 3 x}{2} : 3 x

\frac{2 \cdot 3 x}{2}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

Simplify

2 \cdot 36 0^{\circ} n - 2 \cdot 4 5^{\circ} : 72 0^{\circ} n - 9 0^{\circ}

2 \cdot 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

2 \cdot 36 0^{\circ} n = 72 0^{\circ} n

2 \cdot 36 0^{\circ} n

Multiply the numbers:

2 \cdot 2 = 4

= 72 0^{\circ} n

2 \cdot 4 5^{\circ} = 9 0^{\circ}

2 \cdot 4 5^{\circ}

Multiply fractions:

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

= 9 0^{\circ}

Cancel the common factor:

2

= 9 0^{\circ}

= 72 0^{\circ} n - 9 0^{\circ}

3 x = 72 0^{\circ} n - 9 0^{\circ}

3 x = 72 0^{\circ} n - 9 0^{\circ}

3 x = 72 0^{\circ} n - 9 0^{\circ}

Divide both sides by

3

3 x = 72 0^{\circ} n - 9 0^{\circ}

Divide both sides by

3

\frac{3 x}{3} = \frac{72 0 ^{\circ} n}{3} - \frac{9 0 ^{\circ}}{3}

Simplify

\frac{3 x}{3} = \frac{72 0 ^{\circ} n}{3} - \frac{9 0 ^{\circ}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{72 0 ^{\circ} n}{3} - \frac{9 0 ^{\circ}}{3} : \frac{72 0 ^{\circ} n}{3} - 3 0^{\circ}

\frac{72 0 ^{\circ} n}{3} - \frac{9 0 ^{\circ}}{3}

\frac{9 0 ^{\circ}}{3} = 3 0^{\circ}

\frac{9 0 ^{\circ}}{3}

Apply the fraction rule:

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

= \frac{18 0 ^{\circ}}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= 3 0^{\circ}

= \frac{72 0 ^{\circ} n}{3} - 3 0^{\circ}

x = \frac{72 0 ^{\circ} n}{3} - 3 0^{\circ}

x = \frac{72 0 ^{\circ} n}{3} - 3 0^{\circ}

x = \frac{72 0 ^{\circ} n}{3} - 3 0^{\circ}

Solve

\frac{3 x}{2} + 4 5^{\circ} = 18 0^{\circ} + 36 0^{\circ} n : x = - 3 0^{\circ} + 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

\frac{3 x}{2} + 4 5^{\circ} = 18 0^{\circ} + 36 0^{\circ} n

Move

4 5^{\circ}

to the right side

\frac{3 x}{2} + 4 5^{\circ} = 18 0^{\circ} + 36 0^{\circ} n

Subtract

4 5^{\circ}

from both sides

\frac{3 x}{2} + 4 5^{\circ} - 4 5^{\circ} = 18 0^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Simplify

\frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

\frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Multiply both sides by

2

\frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Multiply both sides by

2

\frac{2 \cdot 3 x}{2} = 36 0^{\circ} + 2 \cdot 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

Simplify

\frac{2 \cdot 3 x}{2} = 36 0^{\circ} + 2 \cdot 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

Simplify

\frac{2 \cdot 3 x}{2} : 3 x

\frac{2 \cdot 3 x}{2}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

Simplify

36 0^{\circ} + 2 \cdot 36 0^{\circ} n - 2 \cdot 4 5^{\circ} : 36 0^{\circ} + 72 0^{\circ} n - 9 0^{\circ}

36 0^{\circ} + 2 \cdot 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

2 \cdot 36 0^{\circ} n = 72 0^{\circ} n

2 \cdot 36 0^{\circ} n

Multiply the numbers:

2 \cdot 2 = 4

= 72 0^{\circ} n

2 \cdot 4 5^{\circ} = 9 0^{\circ}

2 \cdot 4 5^{\circ}

Multiply fractions:

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

= 9 0^{\circ}

Cancel the common factor:

2

= 9 0^{\circ}

= 36 0^{\circ} + 72 0^{\circ} n - 9 0^{\circ}

3 x = 36 0^{\circ} + 72 0^{\circ} n - 9 0^{\circ}

3 x = 36 0^{\circ} + 72 0^{\circ} n - 9 0^{\circ}

3 x = 36 0^{\circ} + 72 0^{\circ} n - 9 0^{\circ}

Divide both sides by

3

3 x = 36 0^{\circ} + 72 0^{\circ} n - 9 0^{\circ}

Divide both sides by

3

\frac{3 x}{3} = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3} - \frac{9 0 ^{\circ}}{3}

Simplify

\frac{3 x}{3} = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3} - \frac{9 0 ^{\circ}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

12 0^{\circ} + \frac{72 0 ^{\circ} n}{3} - \frac{9 0 ^{\circ}}{3} : - 3 0^{\circ} + 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

12 0^{\circ} + \frac{72 0 ^{\circ} n}{3} - \frac{9 0 ^{\circ}}{3}

\frac{9 0 ^{\circ}}{3} = 3 0^{\circ}

\frac{9 0 ^{\circ}}{3}

Apply the fraction rule:

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

= \frac{18 0 ^{\circ}}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= 3 0^{\circ}

= 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3} - 3 0^{\circ}

Group like terms

= - 3 0^{\circ} + 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

x = - 3 0^{\circ} + 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

x = - 3 0^{\circ} + 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

x = - 3 0^{\circ} + 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

x = \frac{72 0 ^{\circ} n}{3} - 3 0^{\circ}, x = - 3 0^{\circ} + 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin((3x)/2+45)=0 ?
The general solution for sin((3x)/2+45)=0 is x=(720n)/3-30,x=-30+120+(720n)/3