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

The general solution for sin(x)(2sin(x)+1)=2 is x=0.89590…+2pin,x=pi-0.89590…+2pin

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

Solution

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

Solution

x = 0.89590 \dots + 2 πn, x = π - 0.89590 \dots + 2 πn

Degrees

x = 51.33171 \dots^{\circ} + 36 0^{\circ} n, x = 128.66828 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

sin (x) = u

u (2 u + 1) = 2

u (2 u + 1) = 2 : u = \frac{- 1 + 17}{4}, u = \frac{- 1 - 17}{4}

u (2 u + 1) = 2

Expand

u (2 u + 1) : 2 u^{2} + u

u (2 u + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = u, b = 2 u, c = 1

= u \cdot 2 u + u \cdot 1

= 2 uu + 1 \cdot u

Simplify

2 uu + 1 \cdot u : 2 u^{2} + u

2 uu + 1 \cdot u

2 uu = 2 u^{2}

2 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 2 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2 u^{2}

1 \cdot u = u

1 \cdot u

Multiply:

1 \cdot u = u

= u

= 2 u^{2} + u

= 2 u^{2} + u

2 u^{2} + u = 2

Move

2

to the left side

2 u^{2} + u = 2

Subtract

2

from both sides

2 u^{2} + u - 2 = 2 - 2

Simplify

2 u^{2} + u - 2 = 0

2 u^{2} + u - 2 = 0

Solve with the quadratic formula

2 u^{2} + u - 2 = 0

Quadratic Equation Formula:

For

a = 2, b = 1, c = - 2

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 2 ( - 2 )}{2 \cdot 2}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 2 ( - 2 )}{2 \cdot 2}

1^{2} - 4 \cdot 2 (- 2) = 17

1^{2} - 4 \cdot 2 (- 2)

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 2 (- 2)

Apply rule

- (- a) = a

= 1 + 4 \cdot 2 \cdot 2

Multiply the numbers:

4 \cdot 2 \cdot 2 = 16

= 1 + 16

Add the numbers:

1 + 16 = 17

= 17

u_{1, 2} = \frac{- 1 \pm 17}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- 1 + 17}{2 \cdot 2}, u_{2} = \frac{- 1 - 17}{2 \cdot 2}

u = \frac{- 1 + 17}{2 \cdot 2} : \frac{- 1 + 17}{4}

\frac{- 1 + 17}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 1 + 17}{4}

u = \frac{- 1 - 17}{2 \cdot 2} : \frac{- 1 - 17}{4}

\frac{- 1 - 17}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 1 - 17}{4}

The solutions to the quadratic equation are:

u = \frac{- 1 + 17}{4}, u = \frac{- 1 - 17}{4}

Substitute back

u = sin (x)

sin (x) = \frac{- 1 + 17}{4}, sin (x) = \frac{- 1 - 17}{4}

sin (x) = \frac{- 1 + 17}{4}, sin (x) = \frac{- 1 - 17}{4}

sin (x) = \frac{- 1 + 17}{4} : x = arcsin (\frac{- 1 + 17}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 17}{4}) + 2 πn

sin (x) = \frac{- 1 + 17}{4}

Apply trig inverse properties

sin (x) = \frac{- 1 + 17}{4}

General solutions for

sin (x) = \frac{- 1 + 17}{4}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{- 1 + 17}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 17}{4}) + 2 πn

x = arcsin (\frac{- 1 + 17}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 17}{4}) + 2 πn

sin (x) = \frac{- 1 - 17}{4} :

No Solution

sin (x) = \frac{- 1 - 17}{4}

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = arcsin (\frac{- 1 + 17}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 17}{4}) + 2 πn

Show solutions in decimal form

x = 0.89590 \dots + 2 πn, x = π - 0.89590 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)(2sin(x)+1)=2 ?
The general solution for sin(x)(2sin(x)+1)=2 is x=0.89590…+2pin,x=pi-0.89590…+2pin