What is the general solution for 2sin(x)+(sqrt(3))/4 = 3/2 sin(x) ?

The general solution for 2sin(x)+(sqrt(3))/4 = 3/2 sin(x) is x=(4pi)/3+2pin,x=(5pi)/3+2pin

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Popular Trigonometry >

2sin(x)+(sqrt(3))/4 = 3/2 sin(x)

Solution

2 sin (x) + \frac{3}{4} = \frac{3}{2} sin (x)

Solution

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Degrees

x = 24 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

2 sin (x) + \frac{3}{4} = \frac{3}{2} sin (x)

Solve by substitution

2 sin (x) + \frac{3}{4} = \frac{3}{2} sin (x)

Let:

sin (x) = u

2 u + \frac{3}{4} = \frac{3}{2} u

2 u + \frac{3}{4} = \frac{3}{2} u : u = - \frac{3}{2}

2 u + \frac{3}{4} = \frac{3}{2} u

Move

\frac{3}{4}

to the right side

2 u + \frac{3}{4} = \frac{3}{2} u

Subtract

\frac{3}{4}

from both sides

2 u + \frac{3}{4} - \frac{3}{4} = \frac{3}{2} u - \frac{3}{4}

Simplify

2 u = \frac{3}{2} u - \frac{3}{4}

2 u = \frac{3}{2} u - \frac{3}{4}

Move

\frac{3}{2} u

to the left side

2 u = \frac{3}{2} u - \frac{3}{4}

Subtract

\frac{3}{2} u

from both sides

2 u - \frac{3}{2} u = \frac{3}{2} u - \frac{3}{4} - \frac{3}{2} u

Simplify

2 u - \frac{3}{2} u = \frac{3}{2} u - \frac{3}{4} - \frac{3}{2} u

Simplify

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

2 u - \frac{3}{2} u

Factor out common term

u

= u (2 - \frac{3}{2})

2 - \frac{3}{2} = \frac{1}{2}

2 - \frac{3}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

2 \cdot 2 - 3 = 1

2 \cdot 2 - 3

Multiply the numbers:

2 \cdot 2 = 4

= 4 - 3

Subtract the numbers:

4 - 3 = 1

= 1

= \frac{1}{2}

= \frac{1}{2} u

Simplify

\frac{3}{2} u - \frac{3}{4} - \frac{3}{2} u : - \frac{3}{4}

\frac{3}{2} u - \frac{3}{4} - \frac{3}{2} u

Add similar elements:

\frac{3}{2} u - \frac{3}{2} u = 0

= - \frac{3}{4}

\frac{1}{2} u = - \frac{3}{4}

\frac{1}{2} u = - \frac{3}{4}

\frac{1}{2} u = - \frac{3}{4}

Multiply both sides by

2

\frac{1}{2} u = - \frac{3}{4}

Multiply both sides by

2

2 \cdot \frac{1}{2} u = 2 (- \frac{3}{4})

Simplify

2 \cdot \frac{1}{2} u = 2 (- \frac{3}{4})

Simplify

2 \cdot \frac{1}{2} u : u

2 \cdot \frac{1}{2} u

Multiply fractions:

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

= \frac{1 \cdot 2}{2} u

Cancel the common factor:

2

= u \cdot 1

Multiply:

u \cdot 1 = u

= u

Simplify

2 (- \frac{3}{4}) : - \frac{3}{2}

2 (- \frac{3}{4})

Remove parentheses:

(- a) = - a

= - 2 \cdot \frac{3}{4}

Multiply fractions:

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

= - \frac{3 \cdot 2}{4}

Cancel the common factor:

2

= - \frac{3}{2}

u = - \frac{3}{2}

u = - \frac{3}{2}

u = - \frac{3}{2}

Substitute back

u = sin (x)

sin (x) = - \frac{3}{2}

sin (x) = - \frac{3}{2}

sin (x) = - \frac{3}{2} : x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

sin (x) = - \frac{3}{2}

General solutions for

sin (x) = - \frac{3}{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}

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Graph

Popular Examples

-2cos^2(θ)+sin(θ)+5=4

Frequently Asked Questions (FAQ)

What is the general solution for 2sin(x)+(sqrt(3))/4 = 3/2 sin(x) ?
The general solution for 2sin(x)+(sqrt(3))/4 = 3/2 sin(x) is x=(4pi)/3+2pin,x=(5pi)/3+2pin