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

The general solution for 2sin(x)+(sqrt(3))/4 = 3/2 is x=0.56272…+2pin,x=pi-0.56272…+2pin

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

Solution

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

Solution

x = 0.56272 \dots + 2 πn, x = π - 0.56272 \dots + 2 πn

Degrees

x = 32.24181 \dots^{\circ} + 36 0^{\circ} n, x = 147.75818 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Move

\frac{3}{4}

to the right side

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

Subtract

\frac{3}{4}

from both sides

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

Simplify

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

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 sin ( x )}{2} : sin (x)

\frac{2 sin ( x )}{2}

Divide the numbers:

\frac{2}{2} = 1

= sin (x)

Simplify

\frac{\frac{3}{2}}{2} - \frac{\frac{3}{4}}{2} : \frac{6 - 3}{8}

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

Apply rule

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

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

Join

\frac{3}{2} - \frac{3}{4} : \frac{6 - 3}{4}

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

Least Common Multiplier of

2, 4 : 4

2, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

2

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

4

For

\frac{3}{2} :

multiply the denominator and numerator by

2

\frac{3}{2} = \frac{3 \cdot 2}{2 \cdot 2} = \frac{6}{4}

= \frac{6}{4} - \frac{3}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{6 - 3}{4}

= \frac{\frac{6 - 3}{4}}{2}

Apply the fraction rule:

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

= \frac{6 - 3}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{6 - 3}{8}

sin (x) = \frac{6 - 3}{8}

sin (x) = \frac{6 - 3}{8}

sin (x) = \frac{6 - 3}{8}

Apply trig inverse properties

sin (x) = \frac{6 - 3}{8}

General solutions for

sin (x) = \frac{6 - 3}{8}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{6 - 3}{8}) + 2 πn, x = π - arcsin (\frac{6 - 3}{8}) + 2 πn

x = arcsin (\frac{6 - 3}{8}) + 2 πn, x = π - arcsin (\frac{6 - 3}{8}) + 2 πn

Show solutions in decimal form

x = 0.56272 \dots + 2 πn, x = π - 0.56272 \dots + 2 πn

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20=27sin(x)-1.5cos(x)

20 = 27 sin (x) - 1.5 cos (x)

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin(x)+(sqrt(3))/4 = 3/2 ?
The general solution for 2sin(x)+(sqrt(3))/4 = 3/2 is x=0.56272…+2pin,x=pi-0.56272…+2pin