What is the general solution for 2sin(x)-cos(x)=0.2 ?

The general solution for 2sin(x)-cos(x)=0.2 is x=-2.76750…+2pin,x=0.55321…+2pin

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2sin(x)-cos(x)=0.2

Solution

2 sin (x) - cos (x) = 0.2

Solution

x = - 2.76750 \dots + 2 πn, x = 0.55321 \dots + 2 πn

Degrees

x = - 158.56649 \dots^{\circ} + 36 0^{\circ} n, x = 31.69659 \dots^{\circ} + 36 0^{\circ} n

Solution steps

2 sin (x) - cos (x) = 0.2

Add

cos (x)

to both sides

2 sin (x) = 0.2 + cos (x)

Square both sides

(2 sin (x))^{2} = (0.2 + cos (x))^{2}

Subtract

(0.2 + cos (x))^{2}

from both sides

4 sin^{2} (x) - 0.04 - 0.4 cos (x) - cos^{2} (x) = 0

Rewrite using trig identities

- 0.04 - cos^{2} (x) - 0.4 cos (x) + 4 sin^{2} (x)

Use the Pythagorean identity:

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

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

= - 0.04 - cos^{2} (x) - 0.4 cos (x) + 4 (1 - cos^{2} (x))

Simplify

- 0.04 - cos^{2} (x) - 0.4 cos (x) + 4 (1 - cos^{2} (x)) : - 5 cos^{2} (x) - 0.4 cos (x) + 3.96

- 0.04 - cos^{2} (x) - 0.4 cos (x) + 4 (1 - cos^{2} (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)

= - 0.04 - cos^{2} (x) - 0.4 cos (x) + 4 - 4 cos^{2} (x)

Simplify

- 0.04 - cos^{2} (x) - 0.4 cos (x) + 4 - 4 cos^{2} (x) : - 5 cos^{2} (x) - 0.4 cos (x) + 3.96

- 0.04 - cos^{2} (x) - 0.4 cos (x) + 4 - 4 cos^{2} (x)

Group like terms

= - cos^{2} (x) - 0.4 cos (x) - 4 cos^{2} (x) - 0.04 + 4

Add similar elements:

- cos^{2} (x) - 4 cos^{2} (x) = - 5 cos^{2} (x)

= - 5 cos^{2} (x) - 0.4 cos (x) - 0.04 + 4

Add/Subtract the numbers:

- 0.04 + 4 = 3.96

= - 5 cos^{2} (x) - 0.4 cos (x) + 3.96

= - 5 cos^{2} (x) - 0.4 cos (x) + 3.96

= - 5 cos^{2} (x) - 0.4 cos (x) + 3.96

3.96 - 0.4 cos (x) - 5 cos^{2} (x) = 0

Solve by substitution

3.96 - 0.4 cos (x) - 5 cos^{2} (x) = 0

Let:

cos (x) = u

3.96 - 0.4 u - 5 u^{2} = 0

3.96 - 0.4 u - 5 u^{2} = 0 : u = - \frac{1 + 4 31}{25}, u = \frac{4 31 - 1}{25}

3.96 - 0.4 u - 5 u^{2} = 0

Multiply both sides by

100

3.96 - 0.4 u - 5 u^{2} = 0

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

3.96 \cdot 100 - 0.4 u \cdot 100 - 5 u^{2} \cdot 100 = 0 \cdot 100

Refine

396 - 40 u - 500 u^{2} = 0

396 - 40 u - 500 u^{2} = 0

Write in the standard form

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

- 500 u^{2} - 40 u + 396 = 0

Solve with the quadratic formula

- 500 u^{2} - 40 u + 396 = 0

Quadratic Equation Formula:

For

a = - 500, b = - 40, c = 396

u_{1, 2} = \frac{- ( - 40 ) \pm ( - 40 ) ^{2} - 4 ( - 500 ) \cdot 396}{2 ( - 500 )}

u_{1, 2} = \frac{- ( - 40 ) \pm ( - 40 ) ^{2} - 4 ( - 500 ) \cdot 396}{2 ( - 500 )}

(- 40)^{2} - 4 (- 500) \cdot 396 = 16031

(- 40)^{2} - 4 (- 500) \cdot 396

Apply rule

- (- a) = a

= (- 40)^{2} + 4 \cdot 500 \cdot 396

Apply exponent rule:

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

n

is even

(- 40)^{2} = 4 0^{2}

= 4 0^{2} + 4 \cdot 500 \cdot 396

Multiply the numbers:

4 \cdot 500 \cdot 396 = 792000

= 4 0^{2} + 792000

4 0^{2} = 1600

= 1600 + 792000

Add the numbers:

1600 + 792000 = 793600

= 793600

Prime factorization of

793600 : 2^{10} \cdot 5^{2} \cdot 31

793600

= 2^{10} \cdot 5^{2} \cdot 31

Apply radical rule:

= 31 5^{2} 2^{10}

Apply radical rule:

2^{10} = 2^{\frac{10}{2}} = 2^{5}

= 2^{5} 31 5^{2}

Apply radical rule:

5^{2} = 5

= 2^{5} \cdot 531

Refine

= 16031

u_{1, 2} = \frac{- ( - 40 ) \pm 160 31}{2 ( - 500 )}

Separate the solutions

u_{1} = \frac{- ( - 40 ) + 160 31}{2 ( - 500 )}, u_{2} = \frac{- ( - 40 ) - 160 31}{2 ( - 500 )}

u = \frac{- ( - 40 ) + 160 31}{2 ( - 500 )} : - \frac{1 + 4 31}{25}

\frac{- ( - 40 ) + 160 31}{2 ( - 500 )}

Remove parentheses:

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

= \frac{40 + 160 31}{- 2 \cdot 500}

Multiply the numbers:

2 \cdot 500 = 1000

= \frac{40 + 160 31}{- 1000}

Apply the fraction rule:

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

= - \frac{40 + 160 31}{1000}

Cancel

\frac{40 + 160 31}{1000} : \frac{1 + 4 31}{25}

\frac{40 + 160 31}{1000}

Factor

40 + 16031 : 40 (1 + 431)

40 + 16031

Rewrite as

= 40 \cdot 1 + 40 \cdot 431

Factor out common term

40

= 40 (1 + 431)

= \frac{40 ( 1 + 4 31 )}{1000}

Cancel the common factor:

40

= \frac{1 + 4 31}{25}

= - \frac{1 + 4 31}{25}

u = \frac{- ( - 40 ) - 160 31}{2 ( - 500 )} : \frac{4 31 - 1}{25}

\frac{- ( - 40 ) - 160 31}{2 ( - 500 )}

Remove parentheses:

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

= \frac{40 - 160 31}{- 2 \cdot 500}

Multiply the numbers:

2 \cdot 500 = 1000

= \frac{40 - 160 31}{- 1000}

Apply the fraction rule:

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

40 - 16031 = - (16031 - 40)

= \frac{160 31 - 40}{1000}

Factor

16031 - 40 : 40 (431 - 1)

16031 - 40

Rewrite as

= 40 \cdot 431 - 40 \cdot 1

Factor out common term

40

= 40 (431 - 1)

= \frac{40 ( 4 31 - 1 )}{1000}

Cancel the common factor:

40

= \frac{4 31 - 1}{25}

The solutions to the quadratic equation are:

u = - \frac{1 + 4 31}{25}, u = \frac{4 31 - 1}{25}

Substitute back

u = cos (x)

cos (x) = - \frac{1 + 4 31}{25}, cos (x) = \frac{4 31 - 1}{25}

cos (x) = - \frac{1 + 4 31}{25}, cos (x) = \frac{4 31 - 1}{25}

cos (x) = - \frac{1 + 4 31}{25} : x = arccos (- \frac{1 + 4 31}{25}) + 2 πn, x = - arccos (- \frac{1 + 4 31}{25}) + 2 πn

cos (x) = - \frac{1 + 4 31}{25}

Apply trig inverse properties

cos (x) = - \frac{1 + 4 31}{25}

General solutions for

cos (x) = - \frac{1 + 4 31}{25}

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

x = arccos (- \frac{1 + 4 31}{25}) + 2 πn, x = - arccos (- \frac{1 + 4 31}{25}) + 2 πn

x = arccos (- \frac{1 + 4 31}{25}) + 2 πn, x = - arccos (- \frac{1 + 4 31}{25}) + 2 πn

cos (x) = \frac{4 31 - 1}{25} : x = arccos (\frac{4 31 - 1}{25}) + 2 πn, x = 2 π - arccos (\frac{4 31 - 1}{25}) + 2 πn

cos (x) = \frac{4 31 - 1}{25}

Apply trig inverse properties

cos (x) = \frac{4 31 - 1}{25}

General solutions for

cos (x) = \frac{4 31 - 1}{25}

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

x = arccos (\frac{4 31 - 1}{25}) + 2 πn, x = 2 π - arccos (\frac{4 31 - 1}{25}) + 2 πn

x = arccos (\frac{4 31 - 1}{25}) + 2 πn, x = 2 π - arccos (\frac{4 31 - 1}{25}) + 2 πn

Combine all the solutions

x = arccos (- \frac{1 + 4 31}{25}) + 2 πn, x = - arccos (- \frac{1 + 4 31}{25}) + 2 πn, x = arccos (\frac{4 31 - 1}{25}) + 2 πn, x = 2 π - arccos (\frac{4 31 - 1}{25}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

2 sin (x) - cos (x) = 0.2

Remove the ones that don't agree with the equation.

Check the solution

arccos (- \frac{1 + 4 31}{25}) + 2 πn :

False

arccos (- \frac{1 + 4 31}{25}) + 2 πn

Plug in

n = 1

arccos (- \frac{1 + 4 31}{25}) + 2 π 1

For

2 sin (x) - cos (x) = 0.2

plug in

x = arccos (- \frac{1 + 4 31}{25}) + 2 π 1

2 sin (arccos (- \frac{1 + 4 31}{25}) + 2 π 1) - cos (arccos (- \frac{1 + 4 31}{25}) + 2 π 1) = 0.2

Refine

1.66168 \dots = 0.2

\Rightarrow False

Check the solution

- arccos (- \frac{1 + 4 31}{25}) + 2 πn :

True

- arccos (- \frac{1 + 4 31}{25}) + 2 πn

Plug in

n = 1

- arccos (- \frac{1 + 4 31}{25}) + 2 π 1

For

2 sin (x) - cos (x) = 0.2

plug in

x = - arccos (- \frac{1 + 4 31}{25}) + 2 π 1

2 sin (- arccos (- \frac{1 + 4 31}{25}) + 2 π 1) - cos (- arccos (- \frac{1 + 4 31}{25}) + 2 π 1) = 0.2

Refine

0.2 = 0.2

\Rightarrow True

Check the solution

arccos (\frac{4 31 - 1}{25}) + 2 πn :

True

arccos (\frac{4 31 - 1}{25}) + 2 πn

Plug in

n = 1

arccos (\frac{4 31 - 1}{25}) + 2 π 1

For

2 sin (x) - cos (x) = 0.2

plug in

x = arccos (\frac{4 31 - 1}{25}) + 2 π 1

2 sin (arccos (\frac{4 31 - 1}{25}) + 2 π 1) - cos (arccos (\frac{4 31 - 1}{25}) + 2 π 1) = 0.2

Refine

0.2 = 0.2

\Rightarrow True

Check the solution

2 π - arccos (\frac{4 31 - 1}{25}) + 2 πn :

False

2 π - arccos (\frac{4 31 - 1}{25}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{4 31 - 1}{25}) + 2 π 1

For

2 sin (x) - cos (x) = 0.2

plug in

x = 2 π - arccos (\frac{4 31 - 1}{25}) + 2 π 1

2 sin (2 π - arccos (\frac{4 31 - 1}{25}) + 2 π 1) - cos (2 π - arccos (\frac{4 31 - 1}{25}) + 2 π 1) = 0.2

Refine

- 1.90168 \dots = 0.2

\Rightarrow False

x = - arccos (- \frac{1 + 4 31}{25}) + 2 πn, x = arccos (\frac{4 31 - 1}{25}) + 2 πn

Show solutions in decimal form

x = - 2.76750 \dots + 2 πn, x = 0.55321 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin(x)-cos(x)=0.2 ?
The general solution for 2sin(x)-cos(x)=0.2 is x=-2.76750…+2pin,x=0.55321…+2pin