What is the general solution for 3sin^2(x)-7sin(x)=6 ?

The general solution for 3sin^2(x)-7sin(x)=6 is x=-0.72972…+2pin,x=pi+0.72972…+2pin

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Popular >

3sin^2(x)-7sin(x)=6

Solution

3 sin^{2} (x) - 7 sin (x) = 6

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) - 7 sin (x) = 6

Solve by substitution

3 sin^{2} (x) - 7 sin (x) = 6

Let:

sin (x) = u

3 u^{2} - 7 u = 6

3 u^{2} - 7 u = 6 : u = 3, u = - \frac{2}{3}

3 u^{2} - 7 u = 6

Move

6

to the left side

3 u^{2} - 7 u = 6

Subtract

6

from both sides

3 u^{2} - 7 u - 6 = 6 - 6

Simplify

3 u^{2} - 7 u - 6 = 0

3 u^{2} - 7 u - 6 = 0

Solve with the quadratic formula

3 u^{2} - 7 u - 6 = 0

Quadratic Equation Formula:

For

a = 3, b = - 7, c = - 6

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 ( - 6 )}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 ( - 6 )}{2 \cdot 3}

(- 7)^{2} - 4 \cdot 3 (- 6) = 11

(- 7)^{2} - 4 \cdot 3 (- 6)

Apply rule

- (- a) = a

= (- 7)^{2} + 4 \cdot 3 \cdot 6

Apply exponent rule:

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

n

is even

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

= 7^{2} + 4 \cdot 3 \cdot 6

Multiply the numbers:

4 \cdot 3 \cdot 6 = 72

= 7^{2} + 72

7^{2} = 49

= 49 + 72

Add the numbers:

49 + 72 = 121

= 121

Factor the number:

121 = 1 1^{2}

= 1 1^{2}

Apply radical rule:

1 1^{2} = 11

= 11

u_{1, 2} = \frac{- ( - 7 ) \pm 11}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- ( - 7 ) + 11}{2 \cdot 3}, u_{2} = \frac{- ( - 7 ) - 11}{2 \cdot 3}

u = \frac{- ( - 7 ) + 11}{2 \cdot 3} : 3

\frac{- ( - 7 ) + 11}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{7 + 11}{2 \cdot 3}

Add the numbers:

7 + 11 = 18

= \frac{18}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{18}{6}

Divide the numbers:

\frac{18}{6} = 3

= 3

u = \frac{- ( - 7 ) - 11}{2 \cdot 3} : - \frac{2}{3}

\frac{- ( - 7 ) - 11}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{7 - 11}{2 \cdot 3}

Subtract the numbers:

7 - 11 = - 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 = 3, u = - \frac{2}{3}

Substitute back

u = sin (x)

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

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

sin (x) = 3 :

No Solution

sin (x) = 3

- 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

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Frequently Asked Questions (FAQ)

What is the general solution for 3sin^2(x)-7sin(x)=6 ?
The general solution for 3sin^2(x)-7sin(x)=6 is x=-0.72972…+2pin,x=pi+0.72972…+2pin