What is the general solution for cos^2(x)+6sin(x)-6=0 ?

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

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cos^2(x)+6sin(x)-6=0

Solution

cos^{2} (x) + 6 sin (x) - 6 = 0

Solution

x = \frac{π}{2} + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

cos^{2} (x) + 6 sin (x) - 6 = 0

Rewrite using trig identities

- 6 + cos^{2} (x) + 6 sin (x)

Use the Pythagorean identity:

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

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

= - 6 + 1 - sin^{2} (x) + 6 sin (x)

Simplify

= 6 sin (x) - sin^{2} (x) - 5

- 5 - sin^{2} (x) + 6 sin (x) = 0

Solve by substitution

- 5 - sin^{2} (x) + 6 sin (x) = 0

Let:

sin (x) = u

- 5 - u^{2} + 6 u = 0

- 5 - u^{2} + 6 u = 0 : u = 1, u = 5

- 5 - u^{2} + 6 u = 0

Write in the standard form

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

- u^{2} + 6 u - 5 = 0

Solve with the quadratic formula

- u^{2} + 6 u - 5 = 0

Quadratic Equation Formula:

For

a = - 1, b = 6, c = - 5

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 1 ) ( - 5 )}{2 ( - 1 )}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 1 ) ( - 5 )}{2 ( - 1 )}

6^{2} - 4 (- 1) (- 5) = 4

6^{2} - 4 (- 1) (- 5)

Apply rule

- (- a) = a

= 6^{2} - 4 \cdot 1 \cdot 5

Multiply the numbers:

4 \cdot 1 \cdot 5 = 20

= 6^{2} - 20

6^{2} = 36

= 36 - 20

Subtract the numbers:

36 - 20 = 16

= 16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

u_{1, 2} = \frac{- 6 \pm 4}{2 ( - 1 )}

Separate the solutions

u_{1} = \frac{- 6 + 4}{2 ( - 1 )}, u_{2} = \frac{- 6 - 4}{2 ( - 1 )}

u = \frac{- 6 + 4}{2 ( - 1 )} : 1

\frac{- 6 + 4}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 6 + 4}{- 2 \cdot 1}

Add/Subtract the numbers:

- 6 + 4 = - 2

= \frac{- 2}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 2}{- 2}

Apply the fraction rule:

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

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

u = \frac{- 6 - 4}{2 ( - 1 )} : 5

\frac{- 6 - 4}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 6 - 4}{- 2 \cdot 1}

Subtract the numbers:

- 6 - 4 = - 10

= \frac{- 10}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 10}{- 2}

Apply the fraction rule:

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

= \frac{10}{2}

Divide the numbers:

\frac{10}{2} = 5

= 5

The solutions to the quadratic equation are:

u = 1, u = 5

Substitute back

u = sin (x)

sin (x) = 1, sin (x) = 5

sin (x) = 1, sin (x) = 5

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 1

General solutions for

sin (x) = 1

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{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

sin (x) = 5 :

No Solution

sin (x) = 5

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = \frac{π}{2} + 2 πn

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Frequently Asked Questions (FAQ)

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