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

The general solution for 4sin^2(x)-7cos(x)=2 is x=1.31811…+2pin,x=2pi-1.31811…+2pin

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

Solution

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

Solution

x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

Degrees

x = 75.52248 \dots^{\circ} + 36 0^{\circ} n, x = 284.47751 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

2

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Multiply the numbers:

4 \cdot 1 = 4

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

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

Add/Subtract the numbers:

- 2 + 4 = 2

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

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

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

Solve by substitution

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

Let:

cos (x) = u

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

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

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

Apply exponent rule:

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

n

is even

(- 7)^{2} = 7^{2}

= 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 )} : - 2

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

Remove parentheses:

(- a) = - a, - (- a) = a

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

Add 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

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

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

Remove parentheses:

(- a) = - a, - (- a) = a

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

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}

The solutions to the quadratic equation are:

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

Substitute back

u = cos (x)

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

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

cos (x) = - 2 :

No Solution

cos (x) = - 2

- 1 \leq cos (x) \leq 1

No Solution

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

cos (x) = \frac{1}{4}

Apply trig inverse properties

cos (x) = \frac{1}{4}

General solutions for

cos (x) = \frac{1}{4}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

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

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

Combine all the solutions

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

Show solutions in decimal form

x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 4sin^2(x)-7cos(x)=2 ?
The general solution for 4sin^2(x)-7cos(x)=2 is x=1.31811…+2pin,x=2pi-1.31811…+2pin