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

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

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

Solution

sin (\frac{x}{2} + π) = \frac{3}{2 3}

Solution

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

Radians

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

Solution steps

sin (\frac{x}{2} + π) = \frac{3}{2 3}

Simplify

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

\frac{3}{2 3}

Apply radical rule:

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

= \frac{3}{2 \cdot 3 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{3}{2}

General solutions for

sin (\frac{x}{2} + π) = \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}

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

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

Solve

\frac{x}{2} + π = \frac{π}{3} + 2 πn : x = \frac{2 π}{3} + 4 πn - 2 π

\frac{x}{2} + π = \frac{π}{3} + 2 πn

Move

π

to the right side

\frac{x}{2} + π = \frac{π}{3} + 2 πn

Subtract

π

from both sides

\frac{x}{2} + π - π = \frac{π}{3} + 2 πn - π

Simplify

\frac{x}{2} = \frac{π}{3} + 2 πn - π

\frac{x}{2} = \frac{π}{3} + 2 πn - π

Multiply both sides by

2

\frac{x}{2} = \frac{π}{3} + 2 πn - π

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{π}{3} + 2 \cdot 2 πn - 2 π

Simplify

\frac{2 x}{2} = 2 \cdot \frac{π}{3} + 2 \cdot 2 πn - 2 π

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

2 \cdot \frac{π}{3} + 2 \cdot 2 πn - 2 π

Multiply

2 \cdot \frac{π}{3} : \frac{2 π}{3}

2 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 2}{3}

= \frac{2 π}{3} + 2 \cdot 2 πn - 2 π

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 π}{3} + 4 πn - 2 π

x = \frac{2 π}{3} + 4 πn - 2 π

x = \frac{2 π}{3} + 4 πn - 2 π

x = \frac{2 π}{3} + 4 πn - 2 π

Solve

\frac{x}{2} + π = \frac{2 π}{3} + 2 πn : x = \frac{4 π}{3} + 4 πn - 2 π

\frac{x}{2} + π = \frac{2 π}{3} + 2 πn

Move

π

to the right side

\frac{x}{2} + π = \frac{2 π}{3} + 2 πn

Subtract

π

from both sides

\frac{x}{2} + π - π = \frac{2 π}{3} + 2 πn - π

Simplify

\frac{x}{2} = \frac{2 π}{3} + 2 πn - π

\frac{x}{2} = \frac{2 π}{3} + 2 πn - π

Multiply both sides by

2

\frac{x}{2} = \frac{2 π}{3} + 2 πn - π

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn - 2 π

Simplify

\frac{2 x}{2} = 2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn - 2 π

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn - 2 π

2 \cdot \frac{2 π}{3} = \frac{4 π}{3}

2 \cdot \frac{2 π}{3}

Multiply fractions:

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

= \frac{2 π 2}{3}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{4 π}{3} + 4 πn - 2 π

x = \frac{4 π}{3} + 4 πn - 2 π

x = \frac{4 π}{3} + 4 πn - 2 π

x = \frac{4 π}{3} + 4 πn - 2 π

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

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cos (b) = \frac{cos ( 8 0 ^{\circ} )}{cos ( 5 0 ^{\circ} )}

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2 sec^{2} (x) tan (x) - 2 sec^{2} (x) = 0

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tan (2 x) = 2 cos (x)

3sec^2(θ)-2tan^2(θ)-4=0

3 sec^{2} (θ) - 2 tan^{2} (θ) - 4 = 0

Frequently Asked Questions (FAQ)

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