What is the general solution for 3sin^2(x)-10sin(x)-8=0 ?

The general solution for 3sin^2(x)-10sin(x)-8=0 is x=-0.72972…+2pin,x=pi+0.72972…+2pin

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Popular >

3sin^2(x)-10sin(x)-8=0

Solution

3 sin^{2} (x) - 10 sin (x) - 8 = 0

Solution

x = - 0.72972 \dots + 2 πn, x = π + 0.72972 \dots + 2 πn

Degrees

x = - 41.81031 \dots^{\circ} + 36 0^{\circ} n, x = 221.81031 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 sin^{2} (x) - 10 sin (x) - 8 = 0

Solve by substitution

3 sin^{2} (x) - 10 sin (x) - 8 = 0

Let:

sin (x) = u

3 u^{2} - 10 u - 8 = 0

3 u^{2} - 10 u - 8 = 0 : u = 4, u = - \frac{2}{3}

3 u^{2} - 10 u - 8 = 0

Solve with the quadratic formula

3 u^{2} - 10 u - 8 = 0

Quadratic Equation Formula:

For

a = 3, b = - 10, c = - 8

u_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 \cdot 3 ( - 8 )}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 \cdot 3 ( - 8 )}{2 \cdot 3}

(- 10)^{2} - 4 \cdot 3 (- 8) = 14

(- 10)^{2} - 4 \cdot 3 (- 8)

Apply rule

- (- a) = a

= (- 10)^{2} + 4 \cdot 3 \cdot 8

Apply exponent rule:

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

n

is even

(- 10)^{2} = 1 0^{2}

= 1 0^{2} + 4 \cdot 3 \cdot 8

Multiply the numbers:

4 \cdot 3 \cdot 8 = 96

= 1 0^{2} + 96

1 0^{2} = 100

= 100 + 96

Add the numbers:

100 + 96 = 196

= 196

Factor the number:

196 = 1 4^{2}

= 1 4^{2}

Apply radical rule:

1 4^{2} = 14

= 14

u_{1, 2} = \frac{- ( - 10 ) \pm 14}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- ( - 10 ) + 14}{2 \cdot 3}, u_{2} = \frac{- ( - 10 ) - 14}{2 \cdot 3}

u = \frac{- ( - 10 ) + 14}{2 \cdot 3} : 4

\frac{- ( - 10 ) + 14}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{10 + 14}{2 \cdot 3}

Add the numbers:

10 + 14 = 24

= \frac{24}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{24}{6}

Divide the numbers:

\frac{24}{6} = 4

= 4

u = \frac{- ( - 10 ) - 14}{2 \cdot 3} : - \frac{2}{3}

\frac{- ( - 10 ) - 14}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{10 - 14}{2 \cdot 3}

Subtract the numbers:

10 - 14 = - 4

= \frac{- 4}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 4}{6}

Apply the fraction rule:

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

= - \frac{4}{6}

Cancel the common factor:

2

= - \frac{2}{3}

The solutions to the quadratic equation are:

u = 4, u = - \frac{2}{3}

Substitute back

u = sin (x)

sin (x) = 4, sin (x) = - \frac{2}{3}

sin (x) = 4, sin (x) = - \frac{2}{3}

sin (x) = 4 :

No Solution

sin (x) = 4

- 1 \leq sin (x) \leq 1

No Solution

sin (x) = - \frac{2}{3} : x = arcsin (- \frac{2}{3}) + 2 πn, x = π + arcsin (\frac{2}{3}) + 2 πn

sin (x) = - \frac{2}{3}

Apply trig inverse properties

sin (x) = - \frac{2}{3}

General solutions for

sin (x) = - \frac{2}{3}

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

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

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

Combine all the solutions

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

Show solutions in decimal form

x = - 0.72972 \dots + 2 πn, x = π + 0.72972 \dots + 2 πn

Popular Examples

2*9.8*L(1-cos(θ))=0.018

2 \cdot 9.8 \cdot L (1 - cos (θ)) = 0.018

cos(7x)=sin(4x+2)

cos (7 x) = sin (4 x + 2)

12cos(β)-5sin(β)=4.7

12 cos (β) - 5 sin (β) = 4.7

cos(x)=(sqrt(8))/3

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

arctan(θ)= 6/8

arctan (θ) = \frac{6}{8}

Frequently Asked Questions (FAQ)

What is the general solution for 3sin^2(x)-10sin(x)-8=0 ?
The general solution for 3sin^2(x)-10sin(x)-8=0 is x=-0.72972…+2pin,x=pi+0.72972…+2pin