What is the general solution for cos^2(x)+3sin(x)+1=0 ?

The general solution for cos^2(x)+3sin(x)+1=0 is x=-0.59626…+2pin,x=pi+0.59626…+2pin

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Popular >

cos^2(x)+3sin(x)+1=0

Solution

cos^{2} (x) + 3 sin (x) + 1 = 0

Solution

x = - 0.59626 \dots + 2 πn, x = π + 0.59626 \dots + 2 πn

Degrees

x = - 34.16325 \dots^{\circ} + 36 0^{\circ} n, x = 214.16325 \dots^{\circ} + 36 0^{\circ} n

Solution steps

cos^{2} (x) + 3 sin (x) + 1 = 0

Rewrite using trig identities

1 + cos^{2} (x) + 3 sin (x)

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Solve by substitution

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

Let:

sin (x) = u

2 - u^{2} + 3 u = 0

2 - u^{2} + 3 u = 0 : u = - \frac{- 3 + 17}{2}, u = \frac{3 + 17}{2}

2 - u^{2} + 3 u = 0

Write in the standard form

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

- u^{2} + 3 u + 2 = 0

Solve with the quadratic formula

- u^{2} + 3 u + 2 = 0

Quadratic Equation Formula:

For

a = - 1, b = 3, c = 2

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

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

3^{2} - 4 (- 1) \cdot 2 = 17

3^{2} - 4 (- 1) \cdot 2

Apply rule

- (- a) = a

= 3^{2} + 4 \cdot 1 \cdot 2

Multiply the numbers:

4 \cdot 1 \cdot 2 = 8

= 3^{2} + 8

3^{2} = 9

= 9 + 8

Add the numbers:

9 + 8 = 17

= 17

u_{1, 2} = \frac{- 3 \pm 17}{2 ( - 1 )}

Separate the solutions

u_{1} = \frac{- 3 + 17}{2 ( - 1 )}, u_{2} = \frac{- 3 - 17}{2 ( - 1 )}

u = \frac{- 3 + 17}{2 ( - 1 )} : - \frac{- 3 + 17}{2}

\frac{- 3 + 17}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 3 + 17}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 3 + 17}{- 2}

Apply the fraction rule:

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

= - \frac{- 3 + 17}{2}

u = \frac{- 3 - 17}{2 ( - 1 )} : \frac{3 + 17}{2}

\frac{- 3 - 17}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 3 - 17}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 3 - 17}{- 2}

Apply the fraction rule:

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

- 3 - 17 = - (3 + 17)

= \frac{3 + 17}{2}

The solutions to the quadratic equation are:

u = - \frac{- 3 + 17}{2}, u = \frac{3 + 17}{2}

Substitute back

u = sin (x)

sin (x) = - \frac{- 3 + 17}{2}, sin (x) = \frac{3 + 17}{2}

sin (x) = - \frac{- 3 + 17}{2}, sin (x) = \frac{3 + 17}{2}

sin (x) = - \frac{- 3 + 17}{2} : x = arcsin (- \frac{- 3 + 17}{2}) + 2 πn, x = π + arcsin (\frac{- 3 + 17}{2}) + 2 πn

sin (x) = - \frac{- 3 + 17}{2}

Apply trig inverse properties

sin (x) = - \frac{- 3 + 17}{2}

General solutions for

sin (x) = - \frac{- 3 + 17}{2}

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

x = arcsin (- \frac{- 3 + 17}{2}) + 2 πn, x = π + arcsin (\frac{- 3 + 17}{2}) + 2 πn

x = arcsin (- \frac{- 3 + 17}{2}) + 2 πn, x = π + arcsin (\frac{- 3 + 17}{2}) + 2 πn

sin (x) = \frac{3 + 17}{2} :

No Solution

sin (x) = \frac{3 + 17}{2}

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = arcsin (- \frac{- 3 + 17}{2}) + 2 πn, x = π + arcsin (\frac{- 3 + 17}{2}) + 2 πn

Show solutions in decimal form

x = - 0.59626 \dots + 2 πn, x = π + 0.59626 \dots + 2 πn

Popular Examples

(a^{0.2})/((cos^2(x))-cos^2(x)-1)=0

\frac{a ^{0.2}}{( cos ^{2} ( x ) ) - cos ^{2} ( x ) - 1} = 0

((2cos(x))/(2-1))((sin(x))/(2+2))=0

(\frac{2 cos ( x )}{2 - 1}) (\frac{sin ( x )}{2 + 2}) = 0

4cos^2(x)+17sin(x)=8

4 cos^{2} (x) + 17 sin (x) = 8

cos^4(x)=1-8sin^2(x)cos^2(x)

cos^{4} (x) = 1 - 8 sin^{2} (x) cos^{2} (x)

tan(x)= 10/18

tan (x) = \frac{10}{18}

Frequently Asked Questions (FAQ)

What is the general solution for cos^2(x)+3sin(x)+1=0 ?
The general solution for cos^2(x)+3sin(x)+1=0 is x=-0.59626…+2pin,x=pi+0.59626…+2pin