What is the general solution for sin(8x)-2cos(4x)=0 ?

The general solution for sin(8x)-2cos(4x)=0 is x=(pi+4pin)/8 ,x=(3pi+4pin)/8

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sin(8x)-2cos(4x)=0

Solution

sin (8 x) - 2 cos (4 x) = 0

Solution

x = \frac{π + 4 πn}{8}, x = \frac{3 π + 4 πn}{8}

Degrees

x = 22. 5^{\circ} + 9 0^{\circ} n, x = 67. 5^{\circ} + 9 0^{\circ} n

Solution steps

sin (8 x) - 2 cos (4 x) = 0

Let:

u = 4 x

sin (2 u) - 2 cos (u) = 0

Rewrite using trig identities

sin (2 u) - 2 cos (u)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= 2 sin (u) cos (u) - 2 cos (u)

- 2 cos (u) + 2 cos (u) sin (u) = 0

Factor

- 2 cos (u) + 2 cos (u) sin (u) : 2 cos (u) (sin (u) - 1)

- 2 cos (u) + 2 cos (u) sin (u)

Factor out common term

2 cos (u)

= 2 cos (u) (- 1 + sin (u))

2 cos (u) (sin (u) - 1) = 0

Solving each part separately

cos (u) = 0 or sin (u) - 1 = 0

cos (u) = 0 : u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

cos (u) = 0

General solutions for

cos (u) = 0

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

sin (u) - 1 = 0 : u = \frac{π}{2} + 2 πn

sin (u) - 1 = 0

Move

1

to the right side

sin (u) - 1 = 0

Add

1

to both sides

sin (u) - 1 + 1 = 0 + 1

Simplify

sin (u) = 1

sin (u) = 1

General solutions for

sin (u) = 1

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}

u = \frac{π}{2} + 2 πn

u = \frac{π}{2} + 2 πn

Combine all the solutions

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

Substitute back

u = 4 x

4 x = \frac{π}{2} + 2 πn : x = \frac{π + 4 πn}{8}

4 x = \frac{π}{2} + 2 πn

Divide both sides by

4

4 x = \frac{π}{2} + 2 πn

Divide both sides by

4

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

Simplify

\frac{4 x}{4} = \frac{\frac{π}{2}}{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}{2}}{4} + \frac{2 πn}{4} : \frac{π + 4 πn}{8}

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

Apply rule

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

= \frac{\frac{π}{2} + 2 πn}{4}

Join

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

\frac{π}{2} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}

= \frac{π}{2} + \frac{2 πn \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{π + 2 πn \cdot 2}{2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π + 4 πn}{2}

= \frac{\frac{π + 4 πn}{2}}{4}

Apply the fraction rule:

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

= \frac{π + 4 πn}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{π + 4 πn}{8}

x = \frac{π + 4 πn}{8}

x = \frac{π + 4 πn}{8}

x = \frac{π + 4 πn}{8}

4 x = \frac{3 π}{2} + 2 πn : x = \frac{3 π + 4 πn}{8}

4 x = \frac{3 π}{2} + 2 πn

Divide both sides by

4

4 x = \frac{3 π}{2} + 2 πn

Divide both sides by

4

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

Simplify

\frac{4 x}{4} = \frac{\frac{3 π}{2}}{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}{2}}{4} + \frac{2 πn}{4} : \frac{3 π + 4 πn}{8}

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

Apply rule

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

= \frac{\frac{3 π}{2} + 2 πn}{4}

Join

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

\frac{3 π}{2} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}

= \frac{3 π}{2} + \frac{2 πn \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{3 π + 2 πn \cdot 2}{2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 π + 4 πn}{2}

= \frac{\frac{3 π + 4 πn}{2}}{4}

Apply the fraction rule:

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

= \frac{3 π + 4 πn}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{3 π + 4 πn}{8}

x = \frac{3 π + 4 πn}{8}

x = \frac{3 π + 4 πn}{8}

x = \frac{3 π + 4 πn}{8}

x = \frac{π + 4 πn}{8}, x = \frac{3 π + 4 πn}{8}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(8x)-2cos(4x)=0 ?
The general solution for sin(8x)-2cos(4x)=0 is x=(pi+4pin)/8 ,x=(3pi+4pin)/8