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Popular >

6/pi-4sin^2(x)=0

Solution

\frac{6}{π} - 4 sin^{2} (x) = 0

Solution

x = 0.76285 \dots + 2 πn, x = π - 0.76285 \dots + 2 πn, x = - 0.76285 \dots + 2 πn, x = π + 0.76285 \dots + 2 πn

+1

Degrees

x = 43.70839 \dots^{\circ} + 36 0^{\circ} n, x = 136.29160 \dots^{\circ} + 36 0^{\circ} n, x = - 43.70839 \dots^{\circ} + 36 0^{\circ} n, x = 223.70839 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{6}{π} - 4 sin^{2} (x) = 0

Solve by substitution

\frac{6}{π} - 4 sin^{2} (x) = 0

Let:

sin (x) = u

\frac{6}{π} - 4 u^{2} = 0

\frac{6}{π} - 4 u^{2} = 0 : u = \frac{3}{2 π}, u = - \frac{3}{2 π}

\frac{6}{π} - 4 u^{2} = 0

Move

\frac{6}{π}

to the right side

\frac{6}{π} - 4 u^{2} = 0

Subtract

\frac{6}{π}

from both sides

\frac{6}{π} - 4 u^{2} - \frac{6}{π} = 0 - \frac{6}{π}

Simplify

- 4 u^{2} = - \frac{6}{π}

- 4 u^{2} = - \frac{6}{π}

Divide both sides by

- 4

- 4 u^{2} = - \frac{6}{π}

Divide both sides by

- 4

\frac{- 4 u ^{2}}{- 4} = \frac{- \frac{6}{π}}{- 4}

Simplify

\frac{- 4 u ^{2}}{- 4} = \frac{- \frac{6}{π}}{- 4}

Simplify

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

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

Apply the fraction rule:

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

= \frac{4 u ^{2}}{4}

Divide the numbers:

\frac{4}{4} = 1

= u^{2}

Simplify

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

\frac{- \frac{6}{π}}{- 4}

Apply the fraction rule:

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

= \frac{\frac{6}{π}}{4}

Apply the fraction rule:

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

= \frac{6}{π 4}

Cancel the common factor:

2

= \frac{3}{2 π}

u^{2} = \frac{3}{2 π}

u^{2} = \frac{3}{2 π}

u^{2} = \frac{3}{2 π}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{3}{2 π}, u = - \frac{3}{2 π}

Substitute back

u = sin (x)

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

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

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

sin (x) = \frac{3}{2 π}

Apply trig inverse properties

sin (x) = \frac{3}{2 π}

General solutions for

sin (x) = \frac{3}{2 π}

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

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

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

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

sin (x) = - \frac{3}{2 π}

Apply trig inverse properties

sin (x) = - \frac{3}{2 π}

General solutions for

sin (x) = - \frac{3}{2 π}

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

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

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

Combine all the solutions

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

Show solutions in decimal form

x = 0.76285 \dots + 2 πn, x = π - 0.76285 \dots + 2 πn, x = - 0.76285 \dots + 2 πn, x = π + 0.76285 \dots + 2 πn

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