What is the general solution for sin(4θ)=4sin(2θ)cos(2θ) ?

The general solution for sin(4θ)=4sin(2θ)cos(2θ) is θ=(pin)/2 ,θ= pi/4+(pin)/2

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sin(4θ)=4sin(2θ)cos(2θ)

Solution

sin (4 θ) = 4 sin (2 θ) cos (2 θ)

Solution

θ = \frac{πn}{2}, θ = \frac{π}{4} + \frac{πn}{2}

Degrees

θ = 0^{\circ} + 9 0^{\circ} n, θ = 4 5^{\circ} + 9 0^{\circ} n

Solution steps

sin (4 θ) = 4 sin (2 θ) cos (2 θ)

Subtract

4 sin (2 θ) cos (2 θ)

from both sides

sin (4 θ) - 4 sin (2 θ) cos (2 θ) = 0

Rewrite using trig identities

sin (4 θ) - 4 cos (2 θ) sin (2 θ)

Use the Double Angle identity:

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

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

= sin (4 θ) - 4 \cdot \frac{sin ( 2 \cdot 2 θ )}{2}

Simplify

sin (4 θ) - 4 \cdot \frac{sin ( 2 \cdot 2 θ )}{2} : - sin (4 θ)

sin (4 θ) - 4 \cdot \frac{sin ( 2 \cdot 2 θ )}{2}

4 \cdot \frac{sin ( 2 \cdot 2 θ )}{2} = 2 sin (4 θ)

4 \cdot \frac{sin ( 2 \cdot 2 θ )}{2}

Multiply the numbers:

2 \cdot 2 = 4

= 4 \cdot \frac{sin ( 4 θ )}{2}

Multiply fractions:

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

= \frac{sin ( 4 θ ) \cdot 4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 sin (4 θ)

= sin (4 θ) - 2 sin (4 θ)

Add similar elements:

sin (4 θ) - 2 sin (4 θ) = - sin (4 θ)

= - sin (4 θ)

= - sin (4 θ)

- sin (4 θ) = 0

Divide both sides by

- 1

- sin (4 θ) = 0

Divide both sides by

- 1

\frac{- sin ( 4 θ )}{- 1} = \frac{0}{- 1}

Simplify

sin (4 θ) = 0

sin (4 θ) = 0

General solutions for

sin (4 θ) = 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}

4 θ = 0 + 2 πn, 4 θ = π + 2 πn

4 θ = 0 + 2 πn, 4 θ = π + 2 πn

Solve

4 θ = 0 + 2 πn : θ = \frac{πn}{2}

4 θ = 0 + 2 πn

0 + 2 πn = 2 πn

4 θ = 2 πn

Divide both sides by

4

4 θ = 2 πn

Divide both sides by

4

\frac{4 θ}{4} = \frac{2 πn}{4}

Simplify

θ = \frac{πn}{2}

θ = \frac{πn}{2}

Solve

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

4 θ = π + 2 πn

Divide both sides by

4

4 θ = π + 2 πn

Divide both sides by

4

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

Simplify

θ = \frac{π}{4} + \frac{πn}{2}

θ = \frac{π}{4} + \frac{πn}{2}

θ = \frac{πn}{2}, θ = \frac{π}{4} + \frac{πn}{2}

Popular Examples

cos(x)=6

cos (x) = 6

6sin(x)-1=4sin(x)

6 sin (x) - 1 = 4 sin (x)

sin(2x)-sin(x)=0,0<= x<2pi

sin (2 x) - sin (x) = 0, 0 \leq x < 2 π

3sin^2(x)-cos^2(x)=1

3 sin^{2} (x) - cos^{2} (x) = 1

2sin^2(θ)+3cos(θ)=3

2 sin^{2} (θ) + 3 cos (θ) = 3

Frequently Asked Questions (FAQ)

What is the general solution for sin(4θ)=4sin(2θ)cos(2θ) ?
The general solution for sin(4θ)=4sin(2θ)cos(2θ) is θ=(pin)/2 ,θ= pi/4+(pin)/2