What is the general solution for 9.8*sin(x)-19.6*cos(x)=4.7 ?

The general solution for 9.8*sin(x)-19.6*cos(x)=4.7 is x=-2.25060…+2pin,x=1.32330…+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

9.8sin(x)-19.6cos(x)=4.7

Solution

9.8 \cdot sin (x) - 19.6 \cdot cos (x) = 4.7

Solution

x = - 2.25060 \dots + 2 πn, x = 1.32330 \dots + 2 πn

Degrees

x = - 128.95007 \dots^{\circ} + 36 0^{\circ} n, x = 75.81997 \dots^{\circ} + 36 0^{\circ} n

Solution steps

9.8 sin (x) - 19.6 cos (x) = 4.7

Add

19.6 cos (x)

to both sides

9.8 sin (x) = 4.7 + 19.6 cos (x)

Square both sides

(9.8 sin (x))^{2} = (4.7 + 19.6 cos (x))^{2}

Subtract

(4.7 + 19.6 cos (x))^{2}

from both sides

96.04 sin^{2} (x) - 22.09 - 184.24 cos (x) - 384.16 cos^{2} (x) = 0

Rewrite using trig identities

- 22.09 - 184.24 cos (x) - 384.16 cos^{2} (x) + 96.04 sin^{2} (x)

Use the Pythagorean identity:

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

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

= - 22.09 - 184.24 cos (x) - 384.16 cos^{2} (x) + 96.04 (1 - cos^{2} (x))

Simplify

- 22.09 - 184.24 cos (x) - 384.16 cos^{2} (x) + 96.04 (1 - cos^{2} (x)) : - 480.2 cos^{2} (x) - 184.24 cos (x) + 73.95

- 22.09 - 184.24 cos (x) - 384.16 cos^{2} (x) + 96.04 (1 - cos^{2} (x))

Expand

96.04 (1 - cos^{2} (x)) : 96.04 - 96.04 cos^{2} (x)

96.04 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 96.04, b = 1, c = cos^{2} (x)

= 96.04 \cdot 1 - 96.04 cos^{2} (x)

= 1 \cdot 96.04 - 96.04 cos^{2} (x)

Multiply the numbers:

1 \cdot 96.04 = 96.04

= 96.04 - 96.04 cos^{2} (x)

= - 22.09 - 184.24 cos (x) - 384.16 cos^{2} (x) + 96.04 - 96.04 cos^{2} (x)

Simplify

- 22.09 - 184.24 cos (x) - 384.16 cos^{2} (x) + 96.04 - 96.04 cos^{2} (x) : - 480.2 cos^{2} (x) - 184.24 cos (x) + 73.95

- 22.09 - 184.24 cos (x) - 384.16 cos^{2} (x) + 96.04 - 96.04 cos^{2} (x)

Group like terms

= - 184.24 cos (x) - 384.16 cos^{2} (x) - 96.04 cos^{2} (x) - 22.09 + 96.04

Add similar elements:

- 384.16 cos^{2} (x) - 96.04 cos^{2} (x) = - 480.2 cos^{2} (x)

= - 184.24 cos (x) - 480.2 cos^{2} (x) - 22.09 + 96.04

Add/Subtract the numbers:

- 22.09 + 96.04 = 73.95

= - 480.2 cos^{2} (x) - 184.24 cos (x) + 73.95

= - 480.2 cos^{2} (x) - 184.24 cos (x) + 73.95

= - 480.2 cos^{2} (x) - 184.24 cos (x) + 73.95

73.95 - 184.24 cos (x) - 480.2 cos^{2} (x) = 0

Solve by substitution

73.95 - 184.24 cos (x) - 480.2 cos^{2} (x) = 0

Let:

cos (x) = u

73.95 - 184.24 u - 480.2 u^{2} = 0

73.95 - 184.24 u - 480.2 u^{2} = 0 : u = - \frac{18424 + 1759875376}{96040}, u = \frac{1759875376 - 18424}{96040}

73.95 - 184.24 u - 480.2 u^{2} = 0

Multiply both sides by

100

73.95 - 184.24 u - 480.2 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

73.95 \cdot 100 - 184.24 u \cdot 100 - 480.2 u^{2} \cdot 100 = 0 \cdot 100

Refine

7395 - 18424 u - 48020 u^{2} = 0

7395 - 18424 u - 48020 u^{2} = 0

Write in the standard form

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

- 48020 u^{2} - 18424 u + 7395 = 0

Solve with the quadratic formula

- 48020 u^{2} - 18424 u + 7395 = 0

Quadratic Equation Formula:

For

a = - 48020, b = - 18424, c = 7395

u_{1, 2} = \frac{- ( - 18424 ) \pm ( - 18424 ) ^{2} - 4 ( - 48020 ) \cdot 7395}{2 ( - 48020 )}

u_{1, 2} = \frac{- ( - 18424 ) \pm ( - 18424 ) ^{2} - 4 ( - 48020 ) \cdot 7395}{2 ( - 48020 )}

(- 18424)^{2} - 4 (- 48020) \cdot 7395 = 1759875376

(- 18424)^{2} - 4 (- 48020) \cdot 7395

Apply rule

- (- a) = a

= (- 18424)^{2} + 4 \cdot 48020 \cdot 7395

Apply exponent rule:

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

n

is even

(- 18424)^{2} = 1842 4^{2}

= 1842 4^{2} + 4 \cdot 48020 \cdot 7395

Multiply the numbers:

4 \cdot 48020 \cdot 7395 = 1420431600

= 1842 4^{2} + 1420431600

1842 4^{2} = 339443776

= 339443776 + 1420431600

Add the numbers:

339443776 + 1420431600 = 1759875376

= 1759875376

u_{1, 2} = \frac{- ( - 18424 ) \pm 1759875376}{2 ( - 48020 )}

Separate the solutions

u_{1} = \frac{- ( - 18424 ) + 1759875376}{2 ( - 48020 )}, u_{2} = \frac{- ( - 18424 ) - 1759875376}{2 ( - 48020 )}

u = \frac{- ( - 18424 ) + 1759875376}{2 ( - 48020 )} : - \frac{18424 + 1759875376}{96040}

\frac{- ( - 18424 ) + 1759875376}{2 ( - 48020 )}

Remove parentheses:

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

= \frac{18424 + 1759875376}{- 2 \cdot 48020}

Multiply the numbers:

2 \cdot 48020 = 96040

= \frac{18424 + 1759875376}{- 96040}

Apply the fraction rule:

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

= - \frac{18424 + 1759875376}{96040}

u = \frac{- ( - 18424 ) - 1759875376}{2 ( - 48020 )} : \frac{1759875376 - 18424}{96040}

\frac{- ( - 18424 ) - 1759875376}{2 ( - 48020 )}

Remove parentheses:

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

= \frac{18424 - 1759875376}{- 2 \cdot 48020}

Multiply the numbers:

2 \cdot 48020 = 96040

= \frac{18424 - 1759875376}{- 96040}

Apply the fraction rule:

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

18424 - 1759875376 = - (1759875376 - 18424)

= \frac{1759875376 - 18424}{96040}

The solutions to the quadratic equation are:

u = - \frac{18424 + 1759875376}{96040}, u = \frac{1759875376 - 18424}{96040}

Substitute back

u = cos (x)

cos (x) = - \frac{18424 + 1759875376}{96040}, cos (x) = \frac{1759875376 - 18424}{96040}

cos (x) = - \frac{18424 + 1759875376}{96040}, cos (x) = \frac{1759875376 - 18424}{96040}

cos (x) = - \frac{18424 + 1759875376}{96040} : x = arccos (- \frac{18424 + 1759875376}{96040}) + 2 πn, x = - arccos (- \frac{18424 + 1759875376}{96040}) + 2 πn

cos (x) = - \frac{18424 + 1759875376}{96040}

Apply trig inverse properties

cos (x) = - \frac{18424 + 1759875376}{96040}

General solutions for

cos (x) = - \frac{18424 + 1759875376}{96040}

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

x = arccos (- \frac{18424 + 1759875376}{96040}) + 2 πn, x = - arccos (- \frac{18424 + 1759875376}{96040}) + 2 πn

x = arccos (- \frac{18424 + 1759875376}{96040}) + 2 πn, x = - arccos (- \frac{18424 + 1759875376}{96040}) + 2 πn

cos (x) = \frac{1759875376 - 18424}{96040} : x = arccos (\frac{1759875376 - 18424}{96040}) + 2 πn, x = 2 π - arccos (\frac{1759875376 - 18424}{96040}) + 2 πn

cos (x) = \frac{1759875376 - 18424}{96040}

Apply trig inverse properties

cos (x) = \frac{1759875376 - 18424}{96040}

General solutions for

cos (x) = \frac{1759875376 - 18424}{96040}

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

x = arccos (\frac{1759875376 - 18424}{96040}) + 2 πn, x = 2 π - arccos (\frac{1759875376 - 18424}{96040}) + 2 πn

x = arccos (\frac{1759875376 - 18424}{96040}) + 2 πn, x = 2 π - arccos (\frac{1759875376 - 18424}{96040}) + 2 πn

Combine all the solutions

x = arccos (- \frac{18424 + 1759875376}{96040}) + 2 πn, x = - arccos (- \frac{18424 + 1759875376}{96040}) + 2 πn, x = arccos (\frac{1759875376 - 18424}{96040}) + 2 πn, x = 2 π - arccos (\frac{1759875376 - 18424}{96040}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

9.8 sin (x) - 19.6 cos (x) = 4.7

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

Check the solution

arccos (- \frac{18424 + 1759875376}{96040}) + 2 πn :

False

arccos (- \frac{18424 + 1759875376}{96040}) + 2 πn

Plug in

n = 1

arccos (- \frac{18424 + 1759875376}{96040}) + 2 π 1

For

9.8 sin (x) - 19.6 cos (x) = 4.7

plug in

x = arccos (- \frac{18424 + 1759875376}{96040}) + 2 π 1

9.8 sin (arccos (- \frac{18424 + 1759875376}{96040}) + 2 π 1) - 19.6 cos (arccos (- \frac{18424 + 1759875376}{96040}) + 2 π 1) = 4.7

Refine

19.94280 \dots = 4.7

\Rightarrow False

Check the solution

- arccos (- \frac{18424 + 1759875376}{96040}) + 2 πn :

True

- arccos (- \frac{18424 + 1759875376}{96040}) + 2 πn

Plug in

n = 1

- arccos (- \frac{18424 + 1759875376}{96040}) + 2 π 1

For

9.8 sin (x) - 19.6 cos (x) = 4.7

plug in

x = - arccos (- \frac{18424 + 1759875376}{96040}) + 2 π 1

9.8 sin (- arccos (- \frac{18424 + 1759875376}{96040}) + 2 π 1) - 19.6 cos (- arccos (- \frac{18424 + 1759875376}{96040}) + 2 π 1) = 4.7

Refine

4.7 = 4.7

\Rightarrow True

Check the solution

arccos (\frac{1759875376 - 18424}{96040}) + 2 πn :

True

arccos (\frac{1759875376 - 18424}{96040}) + 2 πn

Plug in

n = 1

arccos (\frac{1759875376 - 18424}{96040}) + 2 π 1

For

9.8 sin (x) - 19.6 cos (x) = 4.7

plug in

x = arccos (\frac{1759875376 - 18424}{96040}) + 2 π 1

9.8 sin (arccos (\frac{1759875376 - 18424}{96040}) + 2 π 1) - 19.6 cos (arccos (\frac{1759875376 - 18424}{96040}) + 2 π 1) = 4.7

Refine

4.7 = 4.7

\Rightarrow True

Check the solution

2 π - arccos (\frac{1759875376 - 18424}{96040}) + 2 πn :

False

2 π - arccos (\frac{1759875376 - 18424}{96040}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{1759875376 - 18424}{96040}) + 2 π 1

For

9.8 sin (x) - 19.6 cos (x) = 4.7

plug in

x = 2 π - arccos (\frac{1759875376 - 18424}{96040}) + 2 π 1

9.8 sin (2 π - arccos (\frac{1759875376 - 18424}{96040}) + 2 π 1) - 19.6 cos (2 π - arccos (\frac{1759875376 - 18424}{96040}) + 2 π 1) = 4.7

Refine

- 14.30280 \dots = 4.7

\Rightarrow False

x = - arccos (- \frac{18424 + 1759875376}{96040}) + 2 πn, x = arccos (\frac{1759875376 - 18424}{96040}) + 2 πn

Show solutions in decimal form

x = - 2.25060 \dots + 2 πn, x = 1.32330 \dots + 2 πn

Graph

Popular Examples

cos(x)=-sqrt(3)sin(x)

sin(α)= 3/5 ,0<a< pi/2

-(15(0.5cos(x)-sin(x)))/((0.5sin(x)+cos(x))^2)=0

solvefor x,0.33=sin(209.42x)

-2+tan(θ)=-1

Frequently Asked Questions (FAQ)

What is the general solution for 9.8*sin(x)-19.6*cos(x)=4.7 ?
The general solution for 9.8*sin(x)-19.6*cos(x)=4.7 is x=-2.25060…+2pin,x=1.32330…+2pin