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Popular >

10sin^2(4x)=5

Solution

10 sin^{2} (4 x) = 5

Solution

x = \frac{π}{16} + \frac{πn}{2}, x = \frac{3 π}{16} + \frac{πn}{2}, x = \frac{5 π}{16} + \frac{πn}{2}, x = \frac{7 π}{16} + \frac{πn}{2}

Degrees

x = 11.2 5^{\circ} + 9 0^{\circ} n, x = 33.7 5^{\circ} + 9 0^{\circ} n, x = 56.2 5^{\circ} + 9 0^{\circ} n, x = 78.7 5^{\circ} + 9 0^{\circ} n

Solution steps

10 sin^{2} (4 x) = 5

Solve by substitution

10 sin^{2} (4 x) = 5

Let:

sin (4 x) = u

10 u^{2} = 5

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

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

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 (4 x)

sin (4 x) = \frac{1}{2}, sin (4 x) = - \frac{1}{2}

sin (4 x) = \frac{1}{2}, sin (4 x) = - \frac{1}{2}

sin (4 x) = \frac{1}{2} : x = \frac{π}{16} + \frac{πn}{2}, x = \frac{3 π}{16} + \frac{πn}{2}

sin (4 x) = \frac{1}{2}

General solutions for

sin (4 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}

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

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

Solve

4 x = \frac{π}{4} + 2 πn : x = \frac{π}{16} + \frac{πn}{2}

4 x = \frac{π}{4} + 2 πn

Divide both sides by

4

4 x = \frac{π}{4} + 2 πn

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{\frac{π}{4}}{4} + \frac{2 πn}{4} : \frac{π}{16} + \frac{πn}{2}

\frac{\frac{π}{4}}{4} + \frac{2 πn}{4}

\frac{\frac{π}{4}}{4} = \frac{π}{16}

\frac{\frac{π}{4}}{4}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 4}

Multiply the numbers:

4 \cdot 4 = 16

= \frac{π}{16}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{π}{16} + \frac{πn}{2}

x = \frac{π}{16} + \frac{πn}{2}

x = \frac{π}{16} + \frac{πn}{2}

x = \frac{π}{16} + \frac{πn}{2}

Solve

4 x = \frac{3 π}{4} + 2 πn : x = \frac{3 π}{16} + \frac{πn}{2}

4 x = \frac{3 π}{4} + 2 πn

Divide both sides by

4

4 x = \frac{3 π}{4} + 2 πn

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{\frac{3 π}{4}}{4} + \frac{2 πn}{4} : \frac{3 π}{16} + \frac{πn}{2}

\frac{\frac{3 π}{4}}{4} + \frac{2 πn}{4}

\frac{\frac{3 π}{4}}{4} = \frac{3 π}{16}

\frac{\frac{3 π}{4}}{4}

Apply the fraction rule:

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

= \frac{3 π}{4 \cdot 4}

Multiply the numbers:

4 \cdot 4 = 16

= \frac{3 π}{16}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{3 π}{16} + \frac{πn}{2}

x = \frac{3 π}{16} + \frac{πn}{2}

x = \frac{3 π}{16} + \frac{πn}{2}

x = \frac{3 π}{16} + \frac{πn}{2}

x = \frac{π}{16} + \frac{πn}{2}, x = \frac{3 π}{16} + \frac{πn}{2}

sin (4 x) = - \frac{1}{2} : x = \frac{5 π}{16} + \frac{πn}{2}, x = \frac{7 π}{16} + \frac{πn}{2}

sin (4 x) = - \frac{1}{2}

General solutions for

sin (4 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}

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

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

Solve

4 x = \frac{5 π}{4} + 2 πn : x = \frac{5 π}{16} + \frac{πn}{2}

4 x = \frac{5 π}{4} + 2 πn

Divide both sides by

4

4 x = \frac{5 π}{4} + 2 πn

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{\frac{5 π}{4}}{4} + \frac{2 πn}{4} : \frac{5 π}{16} + \frac{πn}{2}

\frac{\frac{5 π}{4}}{4} + \frac{2 πn}{4}

\frac{\frac{5 π}{4}}{4} = \frac{5 π}{16}

\frac{\frac{5 π}{4}}{4}

Apply the fraction rule:

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

= \frac{5 π}{4 \cdot 4}

Multiply the numbers:

4 \cdot 4 = 16

= \frac{5 π}{16}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{5 π}{16} + \frac{πn}{2}

x = \frac{5 π}{16} + \frac{πn}{2}

x = \frac{5 π}{16} + \frac{πn}{2}

x = \frac{5 π}{16} + \frac{πn}{2}

Solve

4 x = \frac{7 π}{4} + 2 πn : x = \frac{7 π}{16} + \frac{πn}{2}

4 x = \frac{7 π}{4} + 2 πn

Divide both sides by

4

4 x = \frac{7 π}{4} + 2 πn

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{\frac{7 π}{4}}{4} + \frac{2 πn}{4} : \frac{7 π}{16} + \frac{πn}{2}

\frac{\frac{7 π}{4}}{4} + \frac{2 πn}{4}

\frac{\frac{7 π}{4}}{4} = \frac{7 π}{16}

\frac{\frac{7 π}{4}}{4}

Apply the fraction rule:

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

= \frac{7 π}{4 \cdot 4}

Multiply the numbers:

4 \cdot 4 = 16

= \frac{7 π}{16}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{7 π}{16} + \frac{πn}{2}

x = \frac{7 π}{16} + \frac{πn}{2}

x = \frac{7 π}{16} + \frac{πn}{2}

x = \frac{7 π}{16} + \frac{πn}{2}

x = \frac{5 π}{16} + \frac{πn}{2}, x = \frac{7 π}{16} + \frac{πn}{2}

Combine all the solutions

x = \frac{π}{16} + \frac{πn}{2}, x = \frac{3 π}{16} + \frac{πn}{2}, x = \frac{5 π}{16} + \frac{πn}{2}, x = \frac{7 π}{16} + \frac{πn}{2}

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