What is the general solution for 4sin^2(x)-7sin(x)-2=0 ?

The general solution for 4sin^2(x)-7sin(x)-2=0 is x=-0.25268…+2pin,x=pi+0.25268…+2pin

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4sin^2(x)-7sin(x)-2=0

Solution

4 sin^{2} (x) - 7 sin (x) - 2 = 0

Solution

x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn

Degrees

x = - 14.47751 \dots^{\circ} + 36 0^{\circ} n, x = 194.47751 \dots^{\circ} + 36 0^{\circ} n

Solution steps

4 sin^{2} (x) - 7 sin (x) - 2 = 0

Solve by substitution

4 sin^{2} (x) - 7 sin (x) - 2 = 0

Let:

sin (x) = u

4 u^{2} - 7 u - 2 = 0

4 u^{2} - 7 u - 2 = 0 : u = 2, u = - \frac{1}{4}

4 u^{2} - 7 u - 2 = 0

Solve with the quadratic formula

4 u^{2} - 7 u - 2 = 0

Quadratic Equation Formula:

For

a = 4, b = - 7, c = - 2

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 4 ( - 2 )}{2 \cdot 4}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 4 ( - 2 )}{2 \cdot 4}

(- 7)^{2} - 4 \cdot 4 (- 2) = 9

(- 7)^{2} - 4 \cdot 4 (- 2)

Apply rule

- (- a) = a

= (- 7)^{2} + 4 \cdot 4 \cdot 2

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 7)^{2} = 7^{2}

= 7^{2} + 4 \cdot 4 \cdot 2

Multiply the numbers:

4 \cdot 4 \cdot 2 = 32

= 7^{2} + 32

7^{2} = 49

= 49 + 32

Add the numbers:

49 + 32 = 81

= 81

Factor the number:

81 = 9^{2}

= 9^{2}

Apply radical rule:

9^{2} = 9

= 9

u_{1, 2} = \frac{- ( - 7 ) \pm 9}{2 \cdot 4}

Separate the solutions

u_{1} = \frac{- ( - 7 ) + 9}{2 \cdot 4}, u_{2} = \frac{- ( - 7 ) - 9}{2 \cdot 4}

u = \frac{- ( - 7 ) + 9}{2 \cdot 4} : 2

\frac{- ( - 7 ) + 9}{2 \cdot 4}

Apply rule

- (- a) = a

= \frac{7 + 9}{2 \cdot 4}

Add the numbers:

7 + 9 = 16

= \frac{16}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{16}{8}

Divide the numbers:

\frac{16}{8} = 2

= 2

u = \frac{- ( - 7 ) - 9}{2 \cdot 4} : - \frac{1}{4}

\frac{- ( - 7 ) - 9}{2 \cdot 4}

Apply rule

- (- a) = a

= \frac{7 - 9}{2 \cdot 4}

Subtract the numbers:

7 - 9 = - 2

= \frac{- 2}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 2}{8}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2}{8}

Cancel the common factor:

2

= - \frac{1}{4}

The solutions to the quadratic equation are:

u = 2, u = - \frac{1}{4}

Substitute back

u = sin (x)

sin (x) = 2, sin (x) = - \frac{1}{4}

sin (x) = 2, sin (x) = - \frac{1}{4}

sin (x) = 2 :

No Solution

sin (x) = 2

- 1 \leq sin (x) \leq 1

No Solution

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

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

Apply trig inverse properties

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

General solutions for

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

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

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

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

Combine all the solutions

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

Show solutions in decimal form

x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn

Popular Examples

cos(x)=-sin(2x)

cos (x) = - sin (2 x)

3cot(x)-sqrt(3)=0

3 cot (x) - 3 = 0

cot(3x)=(sqrt(3))/3 ,0<= x<= 2pi

cot (3 x) = \frac{3}{3}, 0 \leq x \leq 2 π

csc^2(θ)+7csc(θ)+12=0

csc^{2} (θ) + 7 csc (θ) + 12 = 0

2sin(x)+(2-sqrt(2))=sqrt(2)csc(x)

2 sin (x) + (2 - 2) = 2 csc (x)

Frequently Asked Questions (FAQ)

What is the general solution for 4sin^2(x)-7sin(x)-2=0 ?
The general solution for 4sin^2(x)-7sin(x)-2=0 is x=-0.25268…+2pin,x=pi+0.25268…+2pin