What is the general solution for 1/4 sin(8x)=-1/8 ,2cos(8x)=sqrt(3) ?

The general solution for 1/4 sin(8x)=-1/8 ,2cos(8x)=sqrt(3) is No Solution for x\in\mathbb{R}

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1/4 sin(8x)=-1/8 ,2cos(8x)=sqrt(3)

Solution

\frac{1}{4} sin (8 x) = - \frac{1}{8}, 2 cos (8 x) = 3

Solution

No Solution for x \in R

Solution steps

\frac{1}{4} sin (8 x) = - \frac{1}{8}, 2 cos (8 x) = 3

Multiply both sides by

4

\frac{1}{4} sin (8 x) = - \frac{1}{8}

Multiply both sides by

4

4 \cdot \frac{1}{4} sin (8 x) = 4 (- \frac{1}{8})

Simplify

4 \cdot \frac{1}{4} sin (8 x) = 4 (- \frac{1}{8})

Simplify

4 \cdot \frac{1}{4} sin (8 x) : sin (8 x)

4 \cdot \frac{1}{4} sin (8 x)

Multiply fractions:

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

= \frac{1 \cdot 4}{4} sin (8 x)

Cancel the common factor:

4

= sin (8 x) \cdot 1

Multiply:

sin (8 x) \cdot 1 = sin (8 x)

= sin (8 x)

Simplify

4 (- \frac{1}{8}) : - \frac{1}{2}

4 (- \frac{1}{8})

Remove parentheses:

(- a) = - a

= - 4 \cdot \frac{1}{8}

Multiply fractions:

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

= - \frac{1 \cdot 4}{8}

Multiply the numbers:

1 \cdot 4 = 4

= - \frac{4}{8}

Cancel the common factor:

4

= - \frac{1}{2}

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

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

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

General solutions for

sin (8 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}

8 x = \frac{7 π}{6} + 2 πn, 8 x = \frac{11 π}{6} + 2 πn

8 x = \frac{7 π}{6} + 2 πn, 8 x = \frac{11 π}{6} + 2 πn

Solve

8 x = \frac{7 π}{6} + 2 πn : x = \frac{7 π}{48} + \frac{πn}{4}

8 x = \frac{7 π}{6} + 2 πn

Divide both sides by

8

8 x = \frac{7 π}{6} + 2 πn

Divide both sides by

8

\frac{8 x}{8} = \frac{\frac{7 π}{6}}{8} + \frac{2 πn}{8}

Simplify

\frac{8 x}{8} = \frac{\frac{7 π}{6}}{8} + \frac{2 πn}{8}

Simplify

\frac{8 x}{8} : x

\frac{8 x}{8}

Divide the numbers:

\frac{8}{8} = 1

= x

Simplify

\frac{\frac{7 π}{6}}{8} + \frac{2 πn}{8} : \frac{7 π}{48} + \frac{πn}{4}

\frac{\frac{7 π}{6}}{8} + \frac{2 πn}{8}

\frac{\frac{7 π}{6}}{8} = \frac{7 π}{48}

\frac{\frac{7 π}{6}}{8}

Apply the fraction rule:

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

= \frac{7 π}{6 \cdot 8}

Multiply the numbers:

6 \cdot 8 = 48

= \frac{7 π}{48}

\frac{2 πn}{8} = \frac{πn}{4}

\frac{2 πn}{8}

Cancel the common factor:

2

= \frac{πn}{4}

= \frac{7 π}{48} + \frac{πn}{4}

x = \frac{7 π}{48} + \frac{πn}{4}

x = \frac{7 π}{48} + \frac{πn}{4}

x = \frac{7 π}{48} + \frac{πn}{4}

Solve

8 x = \frac{11 π}{6} + 2 πn : x = \frac{11 π}{48} + \frac{πn}{4}

8 x = \frac{11 π}{6} + 2 πn

Divide both sides by

8

8 x = \frac{11 π}{6} + 2 πn

Divide both sides by

8

\frac{8 x}{8} = \frac{\frac{11 π}{6}}{8} + \frac{2 πn}{8}

Simplify

\frac{8 x}{8} = \frac{\frac{11 π}{6}}{8} + \frac{2 πn}{8}

Simplify

\frac{8 x}{8} : x

\frac{8 x}{8}

Divide the numbers:

\frac{8}{8} = 1

= x

Simplify

\frac{\frac{11 π}{6}}{8} + \frac{2 πn}{8} : \frac{11 π}{48} + \frac{πn}{4}

\frac{\frac{11 π}{6}}{8} + \frac{2 πn}{8}

\frac{\frac{11 π}{6}}{8} = \frac{11 π}{48}

\frac{\frac{11 π}{6}}{8}

Apply the fraction rule:

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

= \frac{11 π}{6 \cdot 8}

Multiply the numbers:

6 \cdot 8 = 48

= \frac{11 π}{48}

\frac{2 πn}{8} = \frac{πn}{4}

\frac{2 πn}{8}

Cancel the common factor:

2

= \frac{πn}{4}

= \frac{11 π}{48} + \frac{πn}{4}

x = \frac{11 π}{48} + \frac{πn}{4}

x = \frac{11 π}{48} + \frac{πn}{4}

x = \frac{11 π}{48} + \frac{πn}{4}

x = \frac{7 π}{48} + \frac{πn}{4}, x = \frac{11 π}{48} + \frac{πn}{4}

Solutions for the range

2 cos (8 x) = 3

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for 1/4 sin(8x)=-1/8 ,2cos(8x)=sqrt(3) ?
The general solution for 1/4 sin(8x)=-1/8 ,2cos(8x)=sqrt(3) is No Solution for x\in\mathbb{R}