What is the general solution for 8sin^2(u)+6sin(u)-9=0 ?

The general solution for 8sin^2(u)+6sin(u)-9=0 is u=0.84806…+2pin,u=pi-0.84806…+2pin

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Popular >

8sin^2(u)+6sin(u)-9=0

Solution

8 sin^{2} (u) + 6 sin (u) - 9 = 0

Solution

u = 0.84806 \dots + 2 πn, u = π - 0.84806 \dots + 2 πn

Degrees

u = 48.59037 \dots^{\circ} + 36 0^{\circ} n, u = 131.40962 \dots^{\circ} + 36 0^{\circ} n

Solution steps

8 sin^{2} (u) + 6 sin (u) - 9 = 0

Solve by substitution

8 sin^{2} (u) + 6 sin (u) - 9 = 0

Let:

sin (u) = v

8 v^{2} + 6 v - 9 = 0

8 v^{2} + 6 v - 9 = 0 : v = \frac{3}{4}, v = - \frac{3}{2}

8 v^{2} + 6 v - 9 = 0

Solve with the quadratic formula

8 v^{2} + 6 v - 9 = 0

Quadratic Equation Formula:

For

a = 8, b = 6, c = - 9

v_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 8 ( - 9 )}{2 \cdot 8}

v_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 8 ( - 9 )}{2 \cdot 8}

6^{2} - 4 \cdot 8 (- 9) = 18

6^{2} - 4 \cdot 8 (- 9)

Apply rule

- (- a) = a

= 6^{2} + 4 \cdot 8 \cdot 9

Multiply the numbers:

4 \cdot 8 \cdot 9 = 288

= 6^{2} + 288

6^{2} = 36

= 36 + 288

Add the numbers:

36 + 288 = 324

= 324

Factor the number:

324 = 1 8^{2}

= 1 8^{2}

Apply radical rule:

1 8^{2} = 18

= 18

v_{1, 2} = \frac{- 6 \pm 18}{2 \cdot 8}

Separate the solutions

v_{1} = \frac{- 6 + 18}{2 \cdot 8}, v_{2} = \frac{- 6 - 18}{2 \cdot 8}

v = \frac{- 6 + 18}{2 \cdot 8} : \frac{3}{4}

\frac{- 6 + 18}{2 \cdot 8}

Add/Subtract the numbers:

- 6 + 18 = 12

= \frac{12}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{12}{16}

Cancel the common factor:

4

= \frac{3}{4}

v = \frac{- 6 - 18}{2 \cdot 8} : - \frac{3}{2}

\frac{- 6 - 18}{2 \cdot 8}

Subtract the numbers:

- 6 - 18 = - 24

= \frac{- 24}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{- 24}{16}

Apply the fraction rule:

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

= - \frac{24}{16}

Cancel the common factor:

8

= - \frac{3}{2}

The solutions to the quadratic equation are:

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

Substitute back

v = sin (u)

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

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

sin (u) = \frac{3}{4} : u = arcsin (\frac{3}{4}) + 2 πn, u = π - arcsin (\frac{3}{4}) + 2 πn

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

Apply trig inverse properties

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

General solutions for

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

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

u = arcsin (\frac{3}{4}) + 2 πn, u = π - arcsin (\frac{3}{4}) + 2 πn

u = arcsin (\frac{3}{4}) + 2 πn, u = π - arcsin (\frac{3}{4}) + 2 πn

sin (u) = - \frac{3}{2} :

No Solution

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

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

u = arcsin (\frac{3}{4}) + 2 πn, u = π - arcsin (\frac{3}{4}) + 2 πn

Show solutions in decimal form

u = 0.84806 \dots + 2 πn, u = π - 0.84806 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 8sin^2(u)+6sin(u)-9=0 ?
The general solution for 8sin^2(u)+6sin(u)-9=0 is u=0.84806…+2pin,u=pi-0.84806…+2pin