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

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

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Popular >

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

Solution

3 sin^{2} (x) = 4 sin (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

3 sin^{2} (x) = 4 sin (x)

Solve by substitution

3 sin^{2} (x) = 4 sin (x)

Let:

sin (x) = u

3 u^{2} = 4 u

3 u^{2} = 4 u : u = 0, u = \frac{4}{3}

3 u^{2} = 4 u

Move

4 u

to the left side

3 u^{2} = 4 u

Subtract

4 u

from both sides

3 u^{2} - 4 u = 4 u - 4 u

Simplify

3 u^{2} - 4 u = 0

3 u^{2} - 4 u = 0

Factor

3 u^{2} - 4 u : u (3 u - 4)

3 u^{2} - 4 u

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

u^{2} = uu

= 3 uu - 4 u

Factor out common term

u

= u (3 u - 4)

u (3 u - 4) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u = 0 or 3 u - 4 = 0

Solve

3 u - 4 = 0 : u = \frac{4}{3}

3 u - 4 = 0

Move

4

to the right side

3 u - 4 = 0

Add

4

to both sides

3 u - 4 + 4 = 0 + 4

Simplify

3 u = 4

3 u = 4

Divide both sides by

3

3 u = 4

Divide both sides by

3

\frac{3 u}{3} = \frac{4}{3}

Simplify

u = \frac{4}{3}

u = \frac{4}{3}

The solutions to the quadratic equation are:

u = 0, u = \frac{4}{3}

Substitute back

u = sin (x)

sin (x) = 0, sin (x) = \frac{4}{3}

sin (x) = 0, sin (x) = \frac{4}{3}

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

sin (x) = \frac{4}{3} :

No Solution

sin (x) = \frac{4}{3}

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

Popular Examples

5sin(θ)=4

5 sin (θ) = 4

2cos^2(x)-7cos(x)=0

2 cos^{2} (x) - 7 cos (x) = 0

2cos^2(x)-7cos(x)=3

2 cos^{2} (x) - 7 cos (x) = 3

tan(x)=tan(2x)

tan (x) = tan (2 x)

2cos(x)= 1/2

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

Frequently Asked Questions (FAQ)

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