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

The general solution for 6sin(x)+cos^2(x)=2 is x=0.17242…+2pin,x=pi-0.17242…+2pin

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

Solution

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

Solution

x = 0.17242 \dots + 2 πn, x = π - 0.17242 \dots + 2 πn

Degrees

x = 9.87928 \dots^{\circ} + 36 0^{\circ} n, x = 170.12071 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

2

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Solve by substitution

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

Let:

sin (x) = u

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

- 1 - u^{2} + 6 u = 0 : u = 3 - 22, u = 3 + 22

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

Write in the standard form

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

- u^{2} + 6 u - 1 = 0

Solve with the quadratic formula

- u^{2} + 6 u - 1 = 0

Quadratic Equation Formula:

For

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

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

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

6^{2} - 4 (- 1) (- 1) = 42

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

Apply rule

- (- a) = a

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

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 6^{2} - 4

6^{2} = 36

= 36 - 4

Subtract the numbers:

36 - 4 = 32

= 32

Prime factorization of

32 : 2^{5}

32

32

divides by

232 = 16 \cdot 2

= 2 \cdot 16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

= 2^{5}

= 2^{5}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{4} \cdot 2

Apply radical rule:

= 2 2^{4}

Apply radical rule:

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 2

Refine

= 42

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

Separate the solutions

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

u = \frac{- 6 + 4 2}{2 ( - 1 )} : 3 - 22

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

Remove parentheses:

(- a) = - a

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

Multiply the numbers:

2 \cdot 1 = 2

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

Apply the fraction rule:

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

- 6 + 42 = - (6 - 42)

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

Factor

6 - 42 : 2 (3 - 22)

6 - 42

Rewrite as

= 2 \cdot 3 - 2 \cdot 22

Factor out common term

2

= 2 (3 - 22)

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

Divide the numbers:

\frac{2}{2} = 1

= 3 - 22

u = \frac{- 6 - 4 2}{2 ( - 1 )} : 3 + 22

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

Remove parentheses:

(- a) = - a

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

Multiply the numbers:

2 \cdot 1 = 2

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

Apply the fraction rule:

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

- 6 - 42 = - (6 + 42)

= \frac{6 + 4 2}{2}

Factor

6 + 42 : 2 (3 + 22)

6 + 42

Rewrite as

= 2 \cdot 3 + 2 \cdot 22

Factor out common term

2

= 2 (3 + 22)

= \frac{2 ( 3 + 2 2 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 3 + 22

The solutions to the quadratic equation are:

u = 3 - 22, u = 3 + 22

Substitute back

u = sin (x)

sin (x) = 3 - 22, sin (x) = 3 + 22

sin (x) = 3 - 22, sin (x) = 3 + 22

sin (x) = 3 - 22 : x = arcsin (3 - 22) + 2 πn, x = π - arcsin (3 - 22) + 2 πn

sin (x) = 3 - 22

Apply trig inverse properties

sin (x) = 3 - 22

General solutions for

sin (x) = 3 - 22

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

x = arcsin (3 - 22) + 2 πn, x = π - arcsin (3 - 22) + 2 πn

x = arcsin (3 - 22) + 2 πn, x = π - arcsin (3 - 22) + 2 πn

sin (x) = 3 + 22 :

No Solution

sin (x) = 3 + 22

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = arcsin (3 - 22) + 2 πn, x = π - arcsin (3 - 22) + 2 πn

Show solutions in decimal form

x = 0.17242 \dots + 2 πn, x = π - 0.17242 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 6sin(x)+cos^2(x)=2 ?
The general solution for 6sin(x)+cos^2(x)=2 is x=0.17242…+2pin,x=pi-0.17242…+2pin