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

The general solution for 5cos(x)+2sin(x)=2 is x=-0.80978…+2pin,x= pi/2+2pin

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Popular Trigonometry >

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

Solution

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

Solution

x = - 0.80978 \dots + 2 πn, x = \frac{π}{2} + 2 πn

Degrees

x = - 46.39718 \dots^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

2 sin (x)

from both sides

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

Square both sides

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

Subtract

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

from both sides

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

Rewrite using trig identities

- 4 + 25 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 + 25 (1 - sin^{2} (x)) - 4 sin^{2} (x) + 8 sin (x)

Simplify

- 4 + 25 (1 - sin^{2} (x)) - 4 sin^{2} (x) + 8 sin (x) : 8 sin (x) - 29 sin^{2} (x) + 21

- 4 + 25 (1 - sin^{2} (x)) - 4 sin^{2} (x) + 8 sin (x)

Expand

25 (1 - sin^{2} (x)) : 25 - 25 sin^{2} (x)

25 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 25, b = 1, c = sin^{2} (x)

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

Multiply the numbers:

25 \cdot 1 = 25

= 25 - 25 sin^{2} (x)

= - 4 + 25 - 25 sin^{2} (x) - 4 sin^{2} (x) + 8 sin (x)

Simplify

- 4 + 25 - 25 sin^{2} (x) - 4 sin^{2} (x) + 8 sin (x) : 8 sin (x) - 29 sin^{2} (x) + 21

- 4 + 25 - 25 sin^{2} (x) - 4 sin^{2} (x) + 8 sin (x)

Add similar elements:

- 25 sin^{2} (x) - 4 sin^{2} (x) = - 29 sin^{2} (x)

= - 4 + 25 - 29 sin^{2} (x) + 8 sin (x)

Add/Subtract the numbers:

- 4 + 25 = 21

= 8 sin (x) - 29 sin^{2} (x) + 21

= 8 sin (x) - 29 sin^{2} (x) + 21

= 8 sin (x) - 29 sin^{2} (x) + 21

21 - 29 sin^{2} (x) + 8 sin (x) = 0

Solve by substitution

21 - 29 sin^{2} (x) + 8 sin (x) = 0

Let:

sin (x) = u

21 - 29 u^{2} + 8 u = 0

21 - 29 u^{2} + 8 u = 0 : u = - \frac{21}{29}, u = 1

21 - 29 u^{2} + 8 u = 0

Write in the standard form

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

- 29 u^{2} + 8 u + 21 = 0

Solve with the quadratic formula

- 29 u^{2} + 8 u + 21 = 0

Quadratic Equation Formula:

For

a = - 29, b = 8, c = 21

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 29 ) \cdot 21}{2 ( - 29 )}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 29 ) \cdot 21}{2 ( - 29 )}

8^{2} - 4 (- 29) \cdot 21 = 50

8^{2} - 4 (- 29) \cdot 21

Apply rule

- (- a) = a

= 8^{2} + 4 \cdot 29 \cdot 21

Multiply the numbers:

4 \cdot 29 \cdot 21 = 2436

= 8^{2} + 2436

8^{2} = 64

= 64 + 2436

Add the numbers:

64 + 2436 = 2500

= 2500

Factor the number:

2500 = 5 0^{2}

= 5 0^{2}

Apply radical rule:

5 0^{2} = 50

= 50

u_{1, 2} = \frac{- 8 \pm 50}{2 ( - 29 )}

Separate the solutions

u_{1} = \frac{- 8 + 50}{2 ( - 29 )}, u_{2} = \frac{- 8 - 50}{2 ( - 29 )}

u = \frac{- 8 + 50}{2 ( - 29 )} : - \frac{21}{29}

\frac{- 8 + 50}{2 ( - 29 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 + 50}{- 2 \cdot 29}

Add/Subtract the numbers:

- 8 + 50 = 42

= \frac{42}{- 2 \cdot 29}

Multiply the numbers:

2 \cdot 29 = 58

= \frac{42}{- 58}

Apply the fraction rule:

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

= - \frac{42}{58}

Cancel the common factor:

2

= - \frac{21}{29}

u = \frac{- 8 - 50}{2 ( - 29 )} : 1

\frac{- 8 - 50}{2 ( - 29 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 - 50}{- 2 \cdot 29}

Subtract the numbers:

- 8 - 50 = - 58

= \frac{- 58}{- 2 \cdot 29}

Multiply the numbers:

2 \cdot 29 = 58

= \frac{- 58}{- 58}

Apply the fraction rule:

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

= \frac{58}{58}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

u = - \frac{21}{29}, u = 1

Substitute back

u = sin (x)

sin (x) = - \frac{21}{29}, sin (x) = 1

sin (x) = - \frac{21}{29}, sin (x) = 1

sin (x) = - \frac{21}{29} : x = arcsin (- \frac{21}{29}) + 2 πn, x = π + arcsin (\frac{21}{29}) + 2 πn

sin (x) = - \frac{21}{29}

Apply trig inverse properties

sin (x) = - \frac{21}{29}

General solutions for

sin (x) = - \frac{21}{29}

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

x = arcsin (- \frac{21}{29}) + 2 πn, x = π + arcsin (\frac{21}{29}) + 2 πn

x = arcsin (- \frac{21}{29}) + 2 πn, x = π + arcsin (\frac{21}{29}) + 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{21}{29}) + 2 πn, x = π + arcsin (\frac{21}{29}) + 2 πn, x = \frac{π}{2} + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

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

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

Check the solution

arcsin (- \frac{21}{29}) + 2 πn :

True

arcsin (- \frac{21}{29}) + 2 πn

Plug in

n = 1

arcsin (- \frac{21}{29}) + 2 π 1

For

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

plug in

x = arcsin (- \frac{21}{29}) + 2 π 1

5 cos (arcsin (- \frac{21}{29}) + 2 π 1) + 2 sin (arcsin (- \frac{21}{29}) + 2 π 1) = 2

Refine

2 = 2

\Rightarrow True

Check the solution

π + arcsin (\frac{21}{29}) + 2 πn :

False

π + arcsin (\frac{21}{29}) + 2 πn

Plug in

n = 1

π + arcsin (\frac{21}{29}) + 2 π 1

For

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

plug in

x = π + arcsin (\frac{21}{29}) + 2 π 1

5 cos (π + arcsin (\frac{21}{29}) + 2 π 1) + 2 sin (π + arcsin (\frac{21}{29}) + 2 π 1) = 2

Refine

- 4.89655 \dots = 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

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

plug in

x = \frac{π}{2} + 2 π 1

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

Refine

2 = 2

\Rightarrow True

x = arcsin (- \frac{21}{29}) + 2 πn, x = \frac{π}{2} + 2 πn

Show solutions in decimal form

x = - 0.80978 \dots + 2 πn, x = \frac{π}{2} + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 5cos(x)+2sin(x)=2 ?
The general solution for 5cos(x)+2sin(x)=2 is x=-0.80978…+2pin,x= pi/2+2pin