What is the general solution for 4cos^2(x)+7sin(x)-2=0 ?

The general solution for 4cos^2(x)+7sin(x)-2=0 is x=-0.25268…+2pin,x=pi+0.25268…+2pin

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4cos^2(x)+7sin(x)-2=0

Solution

4 cos^{2} (x) + 7 sin (x) - 2 = 0

Solution

x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn

Degrees

x = - 14.47751 \dots^{\circ} + 36 0^{\circ} n, x = 194.47751 \dots^{\circ} + 36 0^{\circ} n

Solution steps

4 cos^{2} (x) + 7 sin (x) - 2 = 0

Rewrite using trig identities

- 2 + 4 cos^{2} (x) + 7 sin (x)

Use the Pythagorean identity:

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

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

= - 2 + 4 (1 - sin^{2} (x)) + 7 sin (x)

Simplify

- 2 + 4 (1 - sin^{2} (x)) + 7 sin (x) : 7 sin (x) - 4 sin^{2} (x) + 2

- 2 + 4 (1 - sin^{2} (x)) + 7 sin (x)

Expand

4 (1 - sin^{2} (x)) : 4 - 4 sin^{2} (x)

4 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 1, c = sin^{2} (x)

= 4 \cdot 1 - 4 sin^{2} (x)

Multiply the numbers:

4 \cdot 1 = 4

= 4 - 4 sin^{2} (x)

= - 2 + 4 - 4 sin^{2} (x) + 7 sin (x)

Add/Subtract the numbers:

- 2 + 4 = 2

= 7 sin (x) - 4 sin^{2} (x) + 2

= 7 sin (x) - 4 sin^{2} (x) + 2

2 - 4 sin^{2} (x) + 7 sin (x) = 0

Solve by substitution

2 - 4 sin^{2} (x) + 7 sin (x) = 0

Let:

sin (x) = u

2 - 4 u^{2} + 7 u = 0

2 - 4 u^{2} + 7 u = 0 : u = - \frac{1}{4}, u = 2

2 - 4 u^{2} + 7 u = 0

Write in the standard form

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

- 4 u^{2} + 7 u + 2 = 0

Solve with the quadratic formula

- 4 u^{2} + 7 u + 2 = 0

Quadratic Equation Formula:

For

a = - 4, b = 7, c = 2

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 ( - 4 ) \cdot 2}{2 ( - 4 )}

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 ( - 4 ) \cdot 2}{2 ( - 4 )}

7^{2} - 4 (- 4) \cdot 2 = 9

7^{2} - 4 (- 4) \cdot 2

Apply rule

- (- a) = a

= 7^{2} + 4 \cdot 4 \cdot 2

Multiply the numbers:

4 \cdot 4 \cdot 2 = 32

= 7^{2} + 32

7^{2} = 49

= 49 + 32

Add the numbers:

49 + 32 = 81

= 81

Factor the number:

81 = 9^{2}

= 9^{2}

Apply radical rule:

9^{2} = 9

= 9

u_{1, 2} = \frac{- 7 \pm 9}{2 ( - 4 )}

Separate the solutions

u_{1} = \frac{- 7 + 9}{2 ( - 4 )}, u_{2} = \frac{- 7 - 9}{2 ( - 4 )}

u = \frac{- 7 + 9}{2 ( - 4 )} : - \frac{1}{4}

\frac{- 7 + 9}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

= \frac{- 7 + 9}{- 2 \cdot 4}

Add/Subtract the numbers:

- 7 + 9 = 2

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

Multiply the numbers:

2 \cdot 4 = 8

= \frac{2}{- 8}

Apply the fraction rule:

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

= - \frac{2}{8}

Cancel the common factor:

2

= - \frac{1}{4}

u = \frac{- 7 - 9}{2 ( - 4 )} : 2

\frac{- 7 - 9}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

= \frac{- 7 - 9}{- 2 \cdot 4}

Subtract the numbers:

- 7 - 9 = - 16

= \frac{- 16}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 16}{- 8}

Apply the fraction rule:

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

= \frac{16}{8}

Divide the numbers:

\frac{16}{8} = 2

= 2

The solutions to the quadratic equation are:

u = - \frac{1}{4}, u = 2

Substitute back

u = sin (x)

sin (x) = - \frac{1}{4}, sin (x) = 2

sin (x) = - \frac{1}{4}, sin (x) = 2

sin (x) = - \frac{1}{4} : x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

sin (x) = - \frac{1}{4}

Apply trig inverse properties

sin (x) = - \frac{1}{4}

General solutions for

sin (x) = - \frac{1}{4}

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

x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

sin (x) = 2 :

No Solution

sin (x) = 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

Show solutions in decimal form

x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn

Popular Examples

sin(3x)=-(sqrt(2))/2

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

sin(5θ)=0

sin (5 θ) = 0

tanh(x)=0

tanh (x) = 0

cos(x)=cos^2(x)-sin^2(x)

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

sin(x)= 8/15

sin (x) = \frac{8}{15}

Frequently Asked Questions (FAQ)

What is the general solution for 4cos^2(x)+7sin(x)-2=0 ?
The general solution for 4cos^2(x)+7sin(x)-2=0 is x=-0.25268…+2pin,x=pi+0.25268…+2pin