What is the general solution for -2cos^2(x)-9sin(x)-3=0 ?

The general solution for -2cos^2(x)-9sin(x)-3=0 is x=(7pi)/6+2pin,x=(11pi)/6+2pin

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-2cos^2(x)-9sin(x)-3=0

Solution

- 2 cos^{2} (x) - 9 sin (x) - 3 = 0

Solution

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Degrees

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

- 2 cos^{2} (x) - 9 sin (x) - 3 = 0

Rewrite using trig identities

- 3 - 2 cos^{2} (x) - 9 sin (x)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= - 3 - 2 (1 - sin^{2} (x)) - 9 sin (x)

Simplify

- 3 - 2 (1 - sin^{2} (x)) - 9 sin (x) : 2 sin^{2} (x) - 9 sin (x) - 5

- 3 - 2 (1 - sin^{2} (x)) - 9 sin (x)

Expand

- 2 (1 - sin^{2} (x)) : - 2 + 2 sin^{2} (x)

- 2 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 2, b = 1, c = sin^{2} (x)

= - 2 \cdot 1 - (- 2) sin^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 2 \cdot 1 + 2 sin^{2} (x)

Multiply the numbers:

2 \cdot 1 = 2

= - 2 + 2 sin^{2} (x)

= - 3 - 2 + 2 sin^{2} (x) - 9 sin (x)

Subtract the numbers:

- 3 - 2 = - 5

= 2 sin^{2} (x) - 9 sin (x) - 5

= 2 sin^{2} (x) - 9 sin (x) - 5

- 5 + 2 sin^{2} (x) - 9 sin (x) = 0

Solve by substitution

- 5 + 2 sin^{2} (x) - 9 sin (x) = 0

Let:

sin (x) = u

- 5 + 2 u^{2} - 9 u = 0

- 5 + 2 u^{2} - 9 u = 0 : u = 5, u = - \frac{1}{2}

- 5 + 2 u^{2} - 9 u = 0

Write in the standard form

a x^{2} + b x + c = 0

2 u^{2} - 9 u - 5 = 0

Solve with the quadratic formula

2 u^{2} - 9 u - 5 = 0

Quadratic Equation Formula:

For

a = 2, b = - 9, c = - 5

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 \cdot 2 ( - 5 )}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 \cdot 2 ( - 5 )}{2 \cdot 2}

(- 9)^{2} - 4 \cdot 2 (- 5) = 11

(- 9)^{2} - 4 \cdot 2 (- 5)

Apply rule

- (- a) = a

= (- 9)^{2} + 4 \cdot 2 \cdot 5

Apply exponent rule:

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

n

is even

(- 9)^{2} = 9^{2}

= 9^{2} + 4 \cdot 2 \cdot 5

Multiply the numbers:

4 \cdot 2 \cdot 5 = 40

= 9^{2} + 40

9^{2} = 81

= 81 + 40

Add the numbers:

81 + 40 = 121

= 121

Factor the number:

121 = 1 1^{2}

= 1 1^{2}

Apply radical rule:

1 1^{2} = 11

= 11

u_{1, 2} = \frac{- ( - 9 ) \pm 11}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- ( - 9 ) + 11}{2 \cdot 2}, u_{2} = \frac{- ( - 9 ) - 11}{2 \cdot 2}

u = \frac{- ( - 9 ) + 11}{2 \cdot 2} : 5

\frac{- ( - 9 ) + 11}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{9 + 11}{2 \cdot 2}

Add the numbers:

9 + 11 = 20

= \frac{20}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{20}{4}

Divide the numbers:

\frac{20}{4} = 5

= 5

u = \frac{- ( - 9 ) - 11}{2 \cdot 2} : - \frac{1}{2}

\frac{- ( - 9 ) - 11}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{9 - 11}{2 \cdot 2}

Subtract the numbers:

9 - 11 = - 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 2}{4}

Apply the fraction rule:

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

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

The solutions to the quadratic equation are:

u = 5, u = - \frac{1}{2}

Substitute back

u = sin (x)

sin (x) = 5, sin (x) = - \frac{1}{2}

sin (x) = 5, sin (x) = - \frac{1}{2}

sin (x) = 5 :

No Solution

sin (x) = 5

- 1 \leq sin (x) \leq 1

No Solution

sin (x) = - \frac{1}{2} : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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

General solutions for

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

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for -2cos^2(x)-9sin(x)-3=0 ?
The general solution for -2cos^2(x)-9sin(x)-3=0 is x=(7pi)/6+2pin,x=(11pi)/6+2pin