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Popular Trigonometry >

solvefor x,-1/(2y^2)=3sin(x)-1/8

Solution

solve for

x, - \frac{1}{2 y ^{2}} = 3 sin (x) - \frac{1}{8}

Solution

x = arcsin (\frac{- 4 + y ^{2}}{24 y ^{2}}) + 2 πn, x = π + arcsin (- \frac{- 4 + y ^{2}}{24 y ^{2}}) + 2 πn

Solution steps

- \frac{1}{2 y ^{2}} = 3 sin (x) - \frac{1}{8}

Switch sides

3 sin (x) - \frac{1}{8} = - \frac{1}{2 y ^{2}}

Move

\frac{1}{8}

to the right side

3 sin (x) - \frac{1}{8} = - \frac{1}{2 y ^{2}}

Add

\frac{1}{8}

to both sides

3 sin (x) - \frac{1}{8} + \frac{1}{8} = - \frac{1}{2 y ^{2}} + \frac{1}{8}

Simplify

3 sin (x) = - \frac{1}{2 y ^{2}} + \frac{1}{8}

3 sin (x) = - \frac{1}{2 y ^{2}} + \frac{1}{8}

Divide both sides by

3

3 sin (x) = - \frac{1}{2 y ^{2}} + \frac{1}{8}

Divide both sides by

3

\frac{3 sin ( x )}{3} = - \frac{\frac{1}{2 y ^{2}}}{3} + \frac{\frac{1}{8}}{3}

Simplify

\frac{3 sin ( x )}{3} = - \frac{\frac{1}{2 y ^{2}}}{3} + \frac{\frac{1}{8}}{3}

Simplify

\frac{3 sin ( x )}{3} : sin (x)

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

Divide the numbers:

\frac{3}{3} = 1

= sin (x)

Simplify

- \frac{\frac{1}{2 y ^{2}}}{3} + \frac{\frac{1}{8}}{3} : \frac{- 4 + y ^{2}}{24 y ^{2}}

- \frac{\frac{1}{2 y ^{2}}}{3} + \frac{\frac{1}{8}}{3}

Apply rule

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

= \frac{- \frac{1}{2 y ^{2}} + \frac{1}{8}}{3}

Join

- \frac{1}{2 y ^{2}} + \frac{1}{8} : \frac{- 4 + y ^{2}}{8 y ^{2}}

- \frac{1}{2 y ^{2}} + \frac{1}{8}

Least Common Multiplier of

2 y^{2}, 8 : 8 y^{2}

2 y^{2}, 8

Lowest Common Multiplier (LCM)

Least Common Multiplier of

2, 8 : 8

2, 8

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

8 : 2 \cdot 2 \cdot 2

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

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

2

8

= 2 \cdot 2 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 = 8

= 8

Compute an expression comprised of factors that appear either in

2 y^{2}

8

= 8 y^{2}

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

8 y^{2}

For

\frac{1}{2 y ^{2}} :

multiply the denominator and numerator by

4

\frac{1}{2 y ^{2}} = \frac{1 \cdot 4}{2 y ^{2} \cdot 4} = \frac{4}{8 y ^{2}}

For

\frac{1}{8} :

multiply the denominator and numerator by

y^{2}

\frac{1}{8} = \frac{1 \cdot y ^{2}}{8 y ^{2}} = \frac{y ^{2}}{8 y ^{2}}

= - \frac{4}{8 y ^{2}} + \frac{y ^{2}}{8 y ^{2}}

Since the denominators are equal, combine the fractions:

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

= \frac{- 4 + y ^{2}}{8 y ^{2}}

= \frac{\frac{- 4 + y ^{2}}{8 y ^{2}}}{3}

Apply the fraction rule:

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

= \frac{- 4 + y ^{2}}{8 y ^{2} \cdot 3}

Multiply the numbers:

8 \cdot 3 = 24

= \frac{- 4 + y ^{2}}{24 y ^{2}}

sin (x) = \frac{- 4 + y ^{2}}{24 y ^{2}}

sin (x) = \frac{- 4 + y ^{2}}{24 y ^{2}}

sin (x) = \frac{- 4 + y ^{2}}{24 y ^{2}}

Apply trig inverse properties

sin (x) = \frac{- 4 + y ^{2}}{24 y ^{2}}

General solutions for

sin (x) = \frac{- 4 + y ^{2}}{24 y ^{2}}

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

x = arcsin (\frac{- 4 + y ^{2}}{24 y ^{2}}) + 2 πn, x = π + arcsin (- \frac{- 4 + y ^{2}}{24 y ^{2}}) + 2 πn

x = arcsin (\frac{- 4 + y ^{2}}{24 y ^{2}}) + 2 πn, x = π + arcsin (- \frac{- 4 + y ^{2}}{24 y ^{2}}) + 2 πn

Graph

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