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

The general solution for cos(x)+2sin(x)=2 is x=0.64350…+2pin,x= pi/2+2pin

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

Solution

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

Solution

x = 0.64350 \dots + 2 πn, x = \frac{π}{2} + 2 πn

Degrees

x = 36.86989 \dots^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

2 sin (x)

from both sides

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

Square both sides

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

Subtract

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

from both sides

cos^{2} (x) - 4 + 8 sin (x) - 4 sin^{2} (x) = 0

Rewrite using trig identities

- 4 + cos^{2} (x) - 4 sin^{2} (x) + 8 sin (x)

Use the Pythagorean identity:

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

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

= - 4 + 1 - sin^{2} (x) - 4 sin^{2} (x) + 8 sin (x)

Simplify

- 4 + 1 - sin^{2} (x) - 4 sin^{2} (x) + 8 sin (x) : 8 sin (x) - 5 sin^{2} (x) - 3

- 4 + 1 - sin^{2} (x) - 4 sin^{2} (x) + 8 sin (x)

Add similar elements:

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

= - 4 + 1 - 5 sin^{2} (x) + 8 sin (x)

Add/Subtract the numbers:

- 4 + 1 = - 3

= 8 sin (x) - 5 sin^{2} (x) - 3

= 8 sin (x) - 5 sin^{2} (x) - 3

- 3 - 5 sin^{2} (x) + 8 sin (x) = 0

Solve by substitution

- 3 - 5 sin^{2} (x) + 8 sin (x) = 0

Let:

sin (x) = u

- 3 - 5 u^{2} + 8 u = 0

- 3 - 5 u^{2} + 8 u = 0 : u = \frac{3}{5}, u = 1

- 3 - 5 u^{2} + 8 u = 0

Write in the standard form

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

- 5 u^{2} + 8 u - 3 = 0

Solve with the quadratic formula

- 5 u^{2} + 8 u - 3 = 0

Quadratic Equation Formula:

For

a = - 5, b = 8, c = - 3

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 5 ) ( - 3 )}{2 ( - 5 )}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 5 ) ( - 3 )}{2 ( - 5 )}

8^{2} - 4 (- 5) (- 3) = 2

8^{2} - 4 (- 5) (- 3)

Apply rule

- (- a) = a

= 8^{2} - 4 \cdot 5 \cdot 3

Multiply the numbers:

4 \cdot 5 \cdot 3 = 60

= 8^{2} - 60

8^{2} = 64

= 64 - 60

Subtract the numbers:

64 - 60 = 4

= 4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

u_{1, 2} = \frac{- 8 \pm 2}{2 ( - 5 )}

Separate the solutions

u_{1} = \frac{- 8 + 2}{2 ( - 5 )}, u_{2} = \frac{- 8 - 2}{2 ( - 5 )}

u = \frac{- 8 + 2}{2 ( - 5 )} : \frac{3}{5}

\frac{- 8 + 2}{2 ( - 5 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 + 2}{- 2 \cdot 5}

Add/Subtract the numbers:

- 8 + 2 = - 6

= \frac{- 6}{- 2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{- 6}{- 10}

Apply the fraction rule:

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

= \frac{6}{10}

Cancel the common factor:

2

= \frac{3}{5}

u = \frac{- 8 - 2}{2 ( - 5 )} : 1

\frac{- 8 - 2}{2 ( - 5 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 - 2}{- 2 \cdot 5}

Subtract the numbers:

- 8 - 2 = - 10

= \frac{- 10}{- 2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{- 10}{- 10}

Apply the fraction rule:

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

= \frac{10}{10}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

u = \frac{3}{5}, u = 1

Substitute back

u = sin (x)

sin (x) = \frac{3}{5}, sin (x) = 1

sin (x) = \frac{3}{5}, sin (x) = 1

sin (x) = \frac{3}{5} : x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

sin (x) = \frac{3}{5}

Apply trig inverse properties

sin (x) = \frac{3}{5}

General solutions for

sin (x) = \frac{3}{5}

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

x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 1

General solutions for

sin (x) = 1

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Combine all the solutions

x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn, x = \frac{π}{2} + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

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

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

Check the solution

arcsin (\frac{3}{5}) + 2 πn :

True

arcsin (\frac{3}{5}) + 2 πn

Plug in

n = 1

arcsin (\frac{3}{5}) + 2 π 1

For

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

plug in

x = arcsin (\frac{3}{5}) + 2 π 1

cos (arcsin (\frac{3}{5}) + 2 π 1) + 2 sin (arcsin (\frac{3}{5}) + 2 π 1) = 2

Refine

2 = 2

\Rightarrow True

Check the solution

π - arcsin (\frac{3}{5}) + 2 πn :

False

π - arcsin (\frac{3}{5}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{3}{5}) + 2 π 1

For

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

plug in

x = π - arcsin (\frac{3}{5}) + 2 π 1

cos (π - arcsin (\frac{3}{5}) + 2 π 1) + 2 sin (π - arcsin (\frac{3}{5}) + 2 π 1) = 2

Refine

0.4 = 2

\Rightarrow False

Check the solution

\frac{π}{2} + 2 πn :

True

\frac{π}{2} + 2 πn

Plug in

n = 1

\frac{π}{2} + 2 π 1

For

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

plug in

x = \frac{π}{2} + 2 π 1

cos (\frac{π}{2} + 2 π 1) + 2 sin (\frac{π}{2} + 2 π 1) = 2

Refine

2 = 2

\Rightarrow True

x = arcsin (\frac{3}{5}) + 2 πn, x = \frac{π}{2} + 2 πn

Show solutions in decimal form

x = 0.64350 \dots + 2 πn, x = \frac{π}{2} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos(x)+2sin(x)=2 ?
The general solution for cos(x)+2sin(x)=2 is x=0.64350…+2pin,x= pi/2+2pin