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

The general solution for 0=2sin(2x)-sin(x) is x=2pin,x=pi+2pin,x=1.31811…+2pin,x=2pi-1.31811…+2pin

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Popular >

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

Solution

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

Solution

x = 2 πn, x = π + 2 πn, x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 75.52248 \dots^{\circ} + 36 0^{\circ} n, x = 284.47751 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Switch sides

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Simplify

= - sin (x) + 4 sin (x) cos (x)

- sin (x) + 4 cos (x) sin (x) = 0

Factor

- sin (x) + 4 cos (x) sin (x) : sin (x) (4 cos (x) - 1)

- sin (x) + 4 cos (x) sin (x)

Factor out common term

sin (x)

= sin (x) (- 1 + 4 cos (x))

sin (x) (4 cos (x) - 1) = 0

Solving each part separately

sin (x) = 0 or 4 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

4 cos (x) - 1 = 0 : x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

4 cos (x) - 1 = 0

Move

1

to the right side

4 cos (x) - 1 = 0

Add

1

to both sides

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

Simplify

4 cos (x) = 1

4 cos (x) = 1

Divide both sides by

4

4 cos (x) = 1

Divide both sides by

4

\frac{4 cos ( x )}{4} = \frac{1}{4}

Simplify

cos (x) = \frac{1}{4}

cos (x) = \frac{1}{4}

Apply trig inverse properties

cos (x) = \frac{1}{4}

General solutions for

cos (x) = \frac{1}{4}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

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sec^{2} (\frac{x}{3}) = 2

sin(w)=0.79

sin (w) = 0.79

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

tan^{2} (x) - tan (x) = 0, [0 \leq x < 2 π]

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arcsin (x) = 0.5

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Frequently Asked Questions (FAQ)

What is the general solution for 0=2sin(2x)-sin(x) ?
The general solution for 0=2sin(2x)-sin(x) is x=2pin,x=pi+2pin,x=1.31811…+2pin,x=2pi-1.31811…+2pin