What is the general solution for (sin(x))/(sin(2x))= 1/2 ?

The general solution for (sin(x))/(sin(2x))= 1/2 is No Solution for x\in\mathbb{R}

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Popular >

(sin(x))/(sin(2x))= 1/2

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Subtract

\frac{1}{2}

from both sides

\frac{sin ( x )}{sin ( 2 x )} - \frac{1}{2} = 0

Simplify

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

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

Least Common Multiplier of

sin (2 x), 2 : 2 sin (2 x)

sin (2 x), 2

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (2 x)

2

= 2 sin (2 x)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

2 sin (2 x)

For

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

multiply the denominator and numerator by

2

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

For

\frac{1}{2} :

multiply the denominator and numerator by

sin (2 x)

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

Rewrite using trig identities

- sin (2 x) + 2 sin (x)

Use the Double Angle identity:

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

= - 2 sin (x) cos (x) + 2 sin (x)

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

Factor

2 sin (x) - 2 cos (x) sin (x) : - 2 sin (x) (cos (x) - 1)

2 sin (x) - 2 cos (x) sin (x)

Factor out common term

- 2 sin (x)

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

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

Solving each part separately

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

cos (x) - 1 = 0 : x = 2 πn

cos (x) - 1 = 0

Move

1

to the right side

cos (x) - 1 = 0

Add

1

to both sides

cos (x) - 1 + 1 = 0 + 1

Simplify

cos (x) = 1

cos (x) = 1

General solutions for

cos (x) = 1

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}

x = 0 + 2 πn

x = 0 + 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

Combine all the solutions

x = 2 πn, x = π + 2 πn

Since the equation is undefined for:

2 πn, π + 2 πn

No Solution for x \in R

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for (sin(x))/(sin(2x))= 1/2 ?
The general solution for (sin(x))/(sin(2x))= 1/2 is No Solution for x\in\mathbb{R}