What is the general solution for 5sin^2(x)=cos(x) ?

The general solution for 5sin^2(x)=cos(x) is x=0.43944…+2pin,x=2pi-0.43944…+2pin

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Popular >

5sin^2(x)=cos(x)

Solution

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

Solution

x = 0.43944 \dots + 2 πn, x = 2 π - 0.43944 \dots + 2 πn

Degrees

x = 25.17839 \dots^{\circ} + 36 0^{\circ} n, x = 334.82160 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

cos (x)

from both sides

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

Rewrite using trig identities

- cos (x) + 5 sin^{2} (x)

Use the Pythagorean identity:

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

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

= - cos (x) + 5 (1 - cos^{2} (x))

- cos (x) + (1 - cos^{2} (x)) \cdot 5 = 0

Solve by substitution

- cos (x) + (1 - cos^{2} (x)) \cdot 5 = 0

Let:

cos (x) = u

- u + (1 - u^{2}) \cdot 5 = 0

- u + (1 - u^{2}) \cdot 5 = 0 : u = - \frac{1 + 101}{10}, u = \frac{101 - 1}{10}

- u + (1 - u^{2}) \cdot 5 = 0

Expand

- u + (1 - u^{2}) \cdot 5 : - u + 5 - 5 u^{2}

- u + (1 - u^{2}) \cdot 5

= - u + 5 (1 - u^{2})

Expand

5 (1 - u^{2}) : 5 - 5 u^{2}

5 (1 - u^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = 5, b = 1, c = u^{2}

= 5 \cdot 1 - 5 u^{2}

Multiply the numbers:

5 \cdot 1 = 5

= 5 - 5 u^{2}

= - u + 5 - 5 u^{2}

- u + 5 - 5 u^{2} = 0

Write in the standard form

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

- 5 u^{2} - u + 5 = 0

Solve with the quadratic formula

- 5 u^{2} - u + 5 = 0

Quadratic Equation Formula:

For

a = - 5, b = - 1, c = 5

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

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

(- 1)^{2} - 4 (- 5) \cdot 5 = 101

(- 1)^{2} - 4 (- 5) \cdot 5

Apply rule

- (- a) = a

= (- 1)^{2} + 4 \cdot 5 \cdot 5

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

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

n

is even

(- 1)^{2} = 1^{2}

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 5 \cdot 5 = 100

4 \cdot 5 \cdot 5

Multiply the numbers:

4 \cdot 5 \cdot 5 = 100

= 100

= 1 + 100

Add the numbers:

1 + 100 = 101

= 101

u_{1, 2} = \frac{- ( - 1 ) \pm 101}{2 ( - 5 )}

Separate the solutions

u_{1} = \frac{- ( - 1 ) + 101}{2 ( - 5 )}, u_{2} = \frac{- ( - 1 ) - 101}{2 ( - 5 )}

u = \frac{- ( - 1 ) + 101}{2 ( - 5 )} : - \frac{1 + 101}{10}

\frac{- ( - 1 ) + 101}{2 ( - 5 )}

Remove parentheses:

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

= \frac{1 + 101}{- 2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{1 + 101}{- 10}

Apply the fraction rule:

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

= - \frac{1 + 101}{10}

u = \frac{- ( - 1 ) - 101}{2 ( - 5 )} : \frac{101 - 1}{10}

\frac{- ( - 1 ) - 101}{2 ( - 5 )}

Remove parentheses:

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

= \frac{1 - 101}{- 2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{1 - 101}{- 10}

Apply the fraction rule:

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

1 - 101 = - (101 - 1)

= \frac{101 - 1}{10}

The solutions to the quadratic equation are:

u = - \frac{1 + 101}{10}, u = \frac{101 - 1}{10}

Substitute back

u = cos (x)

cos (x) = - \frac{1 + 101}{10}, cos (x) = \frac{101 - 1}{10}

cos (x) = - \frac{1 + 101}{10}, cos (x) = \frac{101 - 1}{10}

cos (x) = - \frac{1 + 101}{10} :

No Solution

cos (x) = - \frac{1 + 101}{10}

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = \frac{101 - 1}{10} : x = arccos (\frac{101 - 1}{10}) + 2 πn, x = 2 π - arccos (\frac{101 - 1}{10}) + 2 πn

cos (x) = \frac{101 - 1}{10}

Apply trig inverse properties

cos (x) = \frac{101 - 1}{10}

General solutions for

cos (x) = \frac{101 - 1}{10}

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

x = arccos (\frac{101 - 1}{10}) + 2 πn, x = 2 π - arccos (\frac{101 - 1}{10}) + 2 πn

x = arccos (\frac{101 - 1}{10}) + 2 πn, x = 2 π - arccos (\frac{101 - 1}{10}) + 2 πn

Combine all the solutions

x = arccos (\frac{101 - 1}{10}) + 2 πn, x = 2 π - arccos (\frac{101 - 1}{10}) + 2 πn

Show solutions in decimal form

x = 0.43944 \dots + 2 πn, x = 2 π - 0.43944 \dots + 2 πn

Popular Examples

cos(x)=sqrt((1+cos(x))/2)

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

cos(x)=(sqrt(1+cos(x)))/2

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

sin(x/2)+(sqrt(3))/2 =0

sin (\frac{x}{2}) + \frac{3}{2} = 0

cos(x)=0.818

cos (x) = 0.818

sin(x)=(150)/(212.6)(sin(115))

sin (x) = \frac{150}{212.6} (sin (11 5^{\circ}))

Frequently Asked Questions (FAQ)

What is the general solution for 5sin^2(x)=cos(x) ?
The general solution for 5sin^2(x)=cos(x) is x=0.43944…+2pin,x=2pi-0.43944…+2pin