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

The general solution for sin^2(x)=4cos(x) is x=1.33247…+2pin,x=2pi-1.33247…+2pin

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

Solution

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

Solution

x = 1.33247 \dots + 2 πn, x = 2 π - 1.33247 \dots + 2 πn

Degrees

x = 76.34541 \dots^{\circ} + 36 0^{\circ} n, x = 283.65458 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

4 cos (x)

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

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

Solve by substitution

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

Let:

cos (x) = u

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

1 - u^{2} - 4 u = 0 : u = - 2 - 5, u = 5 - 2

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

Write in the standard form

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

- u^{2} - 4 u + 1 = 0

Solve with the quadratic formula

- u^{2} - 4 u + 1 = 0

Quadratic Equation Formula:

For

a = - 1, b = - 4, c = 1

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

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

(- 4)^{2} - 4 (- 1) \cdot 1 = 25

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

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

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

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 4^{2} + 4

4^{2} = 16

= 16 + 4

Add the numbers:

16 + 4 = 20

= 20

Prime factorization of

20 : 2^{2} \cdot 5

20

20

divides by

220 = 10 \cdot 2

= 2 \cdot 10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 5

= 2^{2} \cdot 5

= 2^{2} \cdot 5

Apply radical rule:

= 5 2^{2}

Apply radical rule:

2^{2} = 2

= 25

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

Separate the solutions

u_{1} = \frac{- ( - 4 ) + 2 5}{2 ( - 1 )}, u_{2} = \frac{- ( - 4 ) - 2 5}{2 ( - 1 )}

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

\frac{- ( - 4 ) + 2 5}{2 ( - 1 )}

Remove parentheses:

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

= \frac{4 + 2 5}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{4 + 2 5}{- 2}

Apply the fraction rule:

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

= - \frac{4 + 2 5}{2}

Cancel

\frac{4 + 2 5}{2} : 2 + 5

\frac{4 + 2 5}{2}

Factor

4 + 25 : 2 (2 + 5)

4 + 25

Rewrite as

= 2 \cdot 2 + 25

Factor out common term

2

= 2 (2 + 5)

= \frac{2 ( 2 + 5 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2 + 5

= - (2 + 5)

Distribute parentheses

= - (2) - (5)

Apply minus-plus rules

+ (- a) = - a

= - 2 - 5

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

\frac{- ( - 4 ) - 2 5}{2 ( - 1 )}

Remove parentheses:

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

= \frac{4 - 2 5}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{4 - 2 5}{- 2}

Apply the fraction rule:

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

4 - 25 = - (25 - 4)

= \frac{2 5 - 4}{2}

Factor

25 - 4 : 2 (5 - 2)

25 - 4

Rewrite as

= 25 - 2 \cdot 2

Factor out common term

2

= 2 (5 - 2)

= \frac{2 ( 5 - 2 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 5 - 2

The solutions to the quadratic equation are:

u = - 2 - 5, u = 5 - 2

Substitute back

u = cos (x)

cos (x) = - 2 - 5, cos (x) = 5 - 2

cos (x) = - 2 - 5, cos (x) = 5 - 2

cos (x) = - 2 - 5 :

No Solution

cos (x) = - 2 - 5

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = 5 - 2 : x = arccos (5 - 2) + 2 πn, x = 2 π - arccos (5 - 2) + 2 πn

cos (x) = 5 - 2

Apply trig inverse properties

cos (x) = 5 - 2

General solutions for

cos (x) = 5 - 2

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

x = arccos (5 - 2) + 2 πn, x = 2 π - arccos (5 - 2) + 2 πn

x = arccos (5 - 2) + 2 πn, x = 2 π - arccos (5 - 2) + 2 πn

Combine all the solutions

x = arccos (5 - 2) + 2 πn, x = 2 π - arccos (5 - 2) + 2 πn

Show solutions in decimal form

x = 1.33247 \dots + 2 πn, x = 2 π - 1.33247 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)=4cos(x) ?
The general solution for sin^2(x)=4cos(x) is x=1.33247…+2pin,x=2pi-1.33247…+2pin