What is the general solution for 5-10sin^2(x)=0 ?

The general solution for 5-10sin^2(x)=0 is x= pi/4+2pin,x=(3pi)/4+2pin,x=(5pi)/4+2pin,x=(7pi)/4+2pin

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5-10sin^2(x)=0

Solution

5 - 10 sin^{2} (x) = 0

Solution

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

5 - 10 sin^{2} (x) = 0

Solve by substitution

5 - 10 sin^{2} (x) = 0

Let:

sin (x) = u

5 - 10 u^{2} = 0

5 - 10 u^{2} = 0 : u = \frac{1}{2}, u = - \frac{1}{2}

5 - 10 u^{2} = 0

Move

5

to the right side

5 - 10 u^{2} = 0

Subtract

5

from both sides

5 - 10 u^{2} - 5 = 0 - 5

Simplify

- 10 u^{2} = - 5

- 10 u^{2} = - 5

Divide both sides by

- 10

- 10 u^{2} = - 5

Divide both sides by

- 10

\frac{- 10 u ^{2}}{- 10} = \frac{- 5}{- 10}

Simplify

\frac{- 10 u ^{2}}{- 10} = \frac{- 5}{- 10}

Simplify

\frac{- 10 u ^{2}}{- 10} : u^{2}

\frac{- 10 u ^{2}}{- 10}

Apply the fraction rule:

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

= \frac{10 u ^{2}}{10}

Divide the numbers:

\frac{10}{10} = 1

= u^{2}

Simplify

\frac{- 5}{- 10} : \frac{1}{2}

\frac{- 5}{- 10}

Apply the fraction rule:

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

= \frac{5}{10}

Cancel the common factor:

5

= \frac{1}{2}

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

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

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

For

x^{2} = f (a)

the solutions are

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

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

Substitute back

u = sin (x)

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2} : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \frac{1}{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{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

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

sin (x) = - \frac{1}{2} : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sin (x) = - \frac{1}{2}

General solutions for

sin (x) = - \frac{1}{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{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Combine all the solutions

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Popular Examples

sec^2(θ)-2=0

sec^{2} (θ) - 2 = 0

2-sin^2(x)=2cos^2(x/2)

2 - sin^{2} (x) = 2 cos^{2} (\frac{x}{2})

sec(θ/5)=sqrt(10)

sec (\frac{θ}{5}) = 10

(sin(x))^4=((cos(x))^2)/2

(sin (x))^{4} = \frac{( cos ( x ) ) ^{2}}{2}

cos(θ)= 15/17

cos (θ) = \frac{15}{17}

Frequently Asked Questions (FAQ)

What is the general solution for 5-10sin^2(x)=0 ?
The general solution for 5-10sin^2(x)=0 is x= pi/4+2pin,x=(3pi)/4+2pin,x=(5pi)/4+2pin,x=(7pi)/4+2pin