What is the general solution for sin(x)= 8/17 sin(2x) ?

The general solution for sin(x)= 8/17 sin(2x) is x=2pin,x=pi+2pin

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Popular >

sin(x)= 8/17 sin(2x)

Solution

sin (x) = \frac{8}{17} sin (2 x)

Solution

x = 2 πn, x = π + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

sin (x) = \frac{8}{17} sin (2 x)

Subtract

\frac{8}{17} sin (2 x)

from both sides

sin (x) - \frac{8}{17} sin (2 x) = 0

Simplify

sin (x) - \frac{8}{17} sin (2 x) : \frac{17 sin ( x ) - 8 sin ( 2 x )}{17}

sin (x) - \frac{8}{17} sin (2 x)

Multiply

\frac{8}{17} sin (2 x) : \frac{8 sin ( 2 x )}{17}

\frac{8}{17} sin (2 x)

Multiply fractions:

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

= \frac{8 sin ( 2 x )}{17}

= sin (x) - \frac{8 sin ( 2 x )}{17}

Convert element to fraction:

sin (x) = \frac{sin ( x ) 17}{17}

= \frac{sin ( x ) \cdot 17}{17} - \frac{8 sin ( 2 x )}{17}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ( x ) \cdot 17 - 8 sin ( 2 x )}{17}

\frac{17 sin ( x ) - 8 sin ( 2 x )}{17} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

17 sin (x) - 8 sin (2 x) = 0

Rewrite using trig identities

17 sin (x) - 8 sin (2 x)

Use the Double Angle identity:

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

= 17 sin (x) - 8 \cdot 2 sin (x) cos (x)

Simplify

= 17 sin (x) - 16 sin (x) cos (x)

17 sin (x) - 16 cos (x) sin (x) = 0

Factor

17 sin (x) - 16 cos (x) sin (x) : - sin (x) (16 cos (x) - 17)

17 sin (x) - 16 cos (x) sin (x)

Factor out common term

- sin (x)

= - sin (x) (- 17 + 16 cos (x))

- sin (x) (16 cos (x) - 17) = 0

Solving each part separately

sin (x) = 0 or 16 cos (x) - 17 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

16 cos (x) - 17 = 0 :

No Solution

16 cos (x) - 17 = 0

Move

17

to the right side

16 cos (x) - 17 = 0

Add

17

to both sides

16 cos (x) - 17 + 17 = 0 + 17

Simplify

16 cos (x) = 17

16 cos (x) = 17

Divide both sides by

16

16 cos (x) = 17

Divide both sides by

16

\frac{16 cos ( x )}{16} = \frac{17}{16}

Simplify

cos (x) = \frac{17}{16}

cos (x) = \frac{17}{16}

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

Popular Examples

2=1+sin(2x)

2 = 1 + sin (2 x)

cosh(x-1)=2

cosh (x - 1) = 2

tan(x)= 7/25

tan (x) = \frac{7}{25}

tan(x)=-1,0<x<2pi

tan (x) = - 1, 0 < x < 2 π

sec^2(x)-2=tan^2(x),0<= x<= 2pi

sec^{2} (x) - 2 = tan^{2} (x), 0 \leq x \leq 2 π

Frequently Asked Questions (FAQ)

What is the general solution for sin(x)= 8/17 sin(2x) ?
The general solution for sin(x)= 8/17 sin(2x) is x=2pin,x=pi+2pin