What is the general solution for 4sin(2x)=3sin(x) ?

The general solution for 4sin(2x)=3sin(x) is x=2pin,x=pi+2pin,x=1.18639…+2pin,x=2pi-1.18639…+2pin

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Popular Trigonometry >

4sin(2x)=3sin(x)

Solution

4 sin (2 x) = 3 sin (x)

Solution

x = 2 πn, x = π + 2 πn, x = 1.18639 \dots + 2 πn, x = 2 π - 1.18639 \dots + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 67.97568 \dots^{\circ} + 36 0^{\circ} n, x = 292.02431 \dots^{\circ} + 36 0^{\circ} n

Solution steps

4 sin (2 x) = 3 sin (x)

Subtract

3 sin (x)

from both sides

4 sin (2 x) - 3 sin (x) = 0

Rewrite using trig identities

- 3 sin (x) + 4 sin (2 x)

Use the Double Angle identity:

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

= - 3 sin (x) + 4 \cdot 2 sin (x) cos (x)

Simplify

= - 3 sin (x) + 8 sin (x) cos (x)

- 3 sin (x) + 8 cos (x) sin (x) = 0

Factor

- 3 sin (x) + 8 cos (x) sin (x) : sin (x) (8 cos (x) - 3)

- 3 sin (x) + 8 cos (x) sin (x)

Factor out common term

sin (x)

= sin (x) (- 3 + 8 cos (x))

sin (x) (8 cos (x) - 3) = 0

Solving each part separately

sin (x) = 0 or 8 cos (x) - 3 = 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

8 cos (x) - 3 = 0 : x = arccos (\frac{3}{8}) + 2 πn, x = 2 π - arccos (\frac{3}{8}) + 2 πn

8 cos (x) - 3 = 0

Move

3

to the right side

8 cos (x) - 3 = 0

Add

3

to both sides

8 cos (x) - 3 + 3 = 0 + 3

Simplify

8 cos (x) = 3

8 cos (x) = 3

Divide both sides by

8

8 cos (x) = 3

Divide both sides by

8

\frac{8 cos ( x )}{8} = \frac{3}{8}

Simplify

cos (x) = \frac{3}{8}

cos (x) = \frac{3}{8}

Apply trig inverse properties

cos (x) = \frac{3}{8}

General solutions for

cos (x) = \frac{3}{8}

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

x = arccos (\frac{3}{8}) + 2 πn, x = 2 π - arccos (\frac{3}{8}) + 2 πn

x = arccos (\frac{3}{8}) + 2 πn, x = 2 π - arccos (\frac{3}{8}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (\frac{3}{8}) + 2 πn, x = 2 π - arccos (\frac{3}{8}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 1.18639 \dots + 2 πn, x = 2 π - 1.18639 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 4sin(2x)=3sin(x) ?
The general solution for 4sin(2x)=3sin(x) is x=2pin,x=pi+2pin,x=1.18639…+2pin,x=2pi-1.18639…+2pin