What is the general solution for 20=27sin(x)-1.5cos(x) ?

The general solution for 20=27sin(x)-1.5cos(x) is x=2.36461…+2pin,x=0.88797…+2pin

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20=27sin(x)-1.5cos(x)

Solution

20 = 27 sin (x) - 1.5 cos (x)

Solution

x = 2.36461 \dots + 2 πn, x = 0.88797 \dots + 2 πn

Degrees

x = 135.48245 \dots^{\circ} + 36 0^{\circ} n, x = 50.87720 \dots^{\circ} + 36 0^{\circ} n

Solution steps

20 = 27 sin (x) - 1.5 cos (x)

Add

1.5 cos (x)

to both sides

27 sin (x) = 20 + 1.5 cos (x)

Square both sides

(27 sin (x))^{2} = (20 + 1.5 cos (x))^{2}

Subtract

(20 + 1.5 cos (x))^{2}

from both sides

729 sin^{2} (x) - 400 - 60 cos (x) - 2.25 cos^{2} (x) = 0

Rewrite using trig identities

- 400 - 2.25 cos^{2} (x) - 60 cos (x) + 729 sin^{2} (x)

Use the Pythagorean identity:

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

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

= - 400 - 2.25 cos^{2} (x) - 60 cos (x) + 729 (1 - cos^{2} (x))

Simplify

- 400 - 2.25 cos^{2} (x) - 60 cos (x) + 729 (1 - cos^{2} (x)) : - 731.25 cos^{2} (x) - 60 cos (x) + 329

- 400 - 2.25 cos^{2} (x) - 60 cos (x) + 729 (1 - cos^{2} (x))

Expand

729 (1 - cos^{2} (x)) : 729 - 729 cos^{2} (x)

729 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 729, b = 1, c = cos^{2} (x)

= 729 \cdot 1 - 729 cos^{2} (x)

Multiply the numbers:

729 \cdot 1 = 729

= 729 - 729 cos^{2} (x)

= - 400 - 2.25 cos^{2} (x) - 60 cos (x) + 729 - 729 cos^{2} (x)

Simplify

- 400 - 2.25 cos^{2} (x) - 60 cos (x) + 729 - 729 cos^{2} (x) : - 731.25 cos^{2} (x) - 60 cos (x) + 329

- 400 - 2.25 cos^{2} (x) - 60 cos (x) + 729 - 729 cos^{2} (x)

Group like terms

= - 2.25 cos^{2} (x) - 60 cos (x) - 729 cos^{2} (x) - 400 + 729

Add similar elements:

- 2.25 cos^{2} (x) - 729 cos^{2} (x) = - 731.25 cos^{2} (x)

= - 731.25 cos^{2} (x) - 60 cos (x) - 400 + 729

Add/Subtract the numbers:

- 400 + 729 = 329

= - 731.25 cos^{2} (x) - 60 cos (x) + 329

= - 731.25 cos^{2} (x) - 60 cos (x) + 329

= - 731.25 cos^{2} (x) - 60 cos (x) + 329

329 - 60 cos (x) - 731.25 cos^{2} (x) = 0

Solve by substitution

329 - 60 cos (x) - 731.25 cos^{2} (x) = 0

Let:

cos (x) = u

329 - 60 u - 731.25 u^{2} = 0

329 - 60 u - 731.25 u^{2} = 0 : u = - \frac{6000 + 9659250000}{146250}, u = \frac{9659250000 - 6000}{146250}

329 - 60 u - 731.25 u^{2} = 0

Multiply both sides by

100

329 - 60 u - 731.25 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

329 \cdot 100 - 60 u \cdot 100 - 731.25 u^{2} \cdot 100 = 0 \cdot 100

Refine

32900 - 6000 u - 73125 u^{2} = 0

32900 - 6000 u - 73125 u^{2} = 0

Write in the standard form

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

- 73125 u^{2} - 6000 u + 32900 = 0

Solve with the quadratic formula

- 73125 u^{2} - 6000 u + 32900 = 0

Quadratic Equation Formula:

For

a = - 73125, b = - 6000, c = 32900

u_{1, 2} = \frac{- ( - 6000 ) \pm ( - 6000 ) ^{2} - 4 ( - 73125 ) \cdot 32900}{2 ( - 73125 )}

u_{1, 2} = \frac{- ( - 6000 ) \pm ( - 6000 ) ^{2} - 4 ( - 73125 ) \cdot 32900}{2 ( - 73125 )}

(- 6000)^{2} - 4 (- 73125) \cdot 32900 = 9659250000

(- 6000)^{2} - 4 (- 73125) \cdot 32900

Apply rule

- (- a) = a

= (- 6000)^{2} + 4 \cdot 73125 \cdot 32900

Apply exponent rule:

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

n

is even

(- 6000)^{2} = 600 0^{2}

= 600 0^{2} + 4 \cdot 73125 \cdot 32900

Multiply the numbers:

4 \cdot 73125 \cdot 32900 = 9623250000

= 600 0^{2} + 9623250000

600 0^{2} = 36000000

= 36000000 + 9623250000

Add the numbers:

36000000 + 9623250000 = 9659250000

= 9659250000

u_{1, 2} = \frac{- ( - 6000 ) \pm 9659250000}{2 ( - 73125 )}

Separate the solutions

u_{1} = \frac{- ( - 6000 ) + 9659250000}{2 ( - 73125 )}, u_{2} = \frac{- ( - 6000 ) - 9659250000}{2 ( - 73125 )}

u = \frac{- ( - 6000 ) + 9659250000}{2 ( - 73125 )} : - \frac{6000 + 9659250000}{146250}

\frac{- ( - 6000 ) + 9659250000}{2 ( - 73125 )}

Remove parentheses:

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

= \frac{6000 + 9659250000}{- 2 \cdot 73125}

Multiply the numbers:

2 \cdot 73125 = 146250

= \frac{6000 + 9659250000}{- 146250}

Apply the fraction rule:

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

= - \frac{6000 + 9659250000}{146250}

u = \frac{- ( - 6000 ) - 9659250000}{2 ( - 73125 )} : \frac{9659250000 - 6000}{146250}

\frac{- ( - 6000 ) - 9659250000}{2 ( - 73125 )}

Remove parentheses:

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

= \frac{6000 - 9659250000}{- 2 \cdot 73125}

Multiply the numbers:

2 \cdot 73125 = 146250

= \frac{6000 - 9659250000}{- 146250}

Apply the fraction rule:

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

6000 - 9659250000 = - (9659250000 - 6000)

= \frac{9659250000 - 6000}{146250}

The solutions to the quadratic equation are:

u = - \frac{6000 + 9659250000}{146250}, u = \frac{9659250000 - 6000}{146250}

Substitute back

u = cos (x)

cos (x) = - \frac{6000 + 9659250000}{146250}, cos (x) = \frac{9659250000 - 6000}{146250}

cos (x) = - \frac{6000 + 9659250000}{146250}, cos (x) = \frac{9659250000 - 6000}{146250}

cos (x) = - \frac{6000 + 9659250000}{146250} : x = arccos (- \frac{6000 + 9659250000}{146250}) + 2 πn, x = - arccos (- \frac{6000 + 9659250000}{146250}) + 2 πn

cos (x) = - \frac{6000 + 9659250000}{146250}

Apply trig inverse properties

cos (x) = - \frac{6000 + 9659250000}{146250}

General solutions for

cos (x) = - \frac{6000 + 9659250000}{146250}

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

x = arccos (- \frac{6000 + 9659250000}{146250}) + 2 πn, x = - arccos (- \frac{6000 + 9659250000}{146250}) + 2 πn

x = arccos (- \frac{6000 + 9659250000}{146250}) + 2 πn, x = - arccos (- \frac{6000 + 9659250000}{146250}) + 2 πn

cos (x) = \frac{9659250000 - 6000}{146250} : x = arccos (\frac{9659250000 - 6000}{146250}) + 2 πn, x = 2 π - arccos (\frac{9659250000 - 6000}{146250}) + 2 πn

cos (x) = \frac{9659250000 - 6000}{146250}

Apply trig inverse properties

cos (x) = \frac{9659250000 - 6000}{146250}

General solutions for

cos (x) = \frac{9659250000 - 6000}{146250}

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

x = arccos (\frac{9659250000 - 6000}{146250}) + 2 πn, x = 2 π - arccos (\frac{9659250000 - 6000}{146250}) + 2 πn

x = arccos (\frac{9659250000 - 6000}{146250}) + 2 πn, x = 2 π - arccos (\frac{9659250000 - 6000}{146250}) + 2 πn

Combine all the solutions

x = arccos (- \frac{6000 + 9659250000}{146250}) + 2 πn, x = - arccos (- \frac{6000 + 9659250000}{146250}) + 2 πn, x = arccos (\frac{9659250000 - 6000}{146250}) + 2 πn, x = 2 π - arccos (\frac{9659250000 - 6000}{146250}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

27 sin (x) - 1.5 cos (x) = 20

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

Check the solution

arccos (- \frac{6000 + 9659250000}{146250}) + 2 πn :

True

arccos (- \frac{6000 + 9659250000}{146250}) + 2 πn

Plug in

n = 1

arccos (- \frac{6000 + 9659250000}{146250}) + 2 π 1

For

27 sin (x) - 1.5 cos (x) = 20

plug in

x = arccos (- \frac{6000 + 9659250000}{146250}) + 2 π 1

27 sin (arccos (- \frac{6000 + 9659250000}{146250}) + 2 π 1) - 1.5 cos (arccos (- \frac{6000 + 9659250000}{146250}) + 2 π 1) = 20

Refine

20 = 20

\Rightarrow True

Check the solution

- arccos (- \frac{6000 + 9659250000}{146250}) + 2 πn :

False

- arccos (- \frac{6000 + 9659250000}{146250}) + 2 πn

Plug in

n = 1

- arccos (- \frac{6000 + 9659250000}{146250}) + 2 π 1

For

27 sin (x) - 1.5 cos (x) = 20

plug in

x = - arccos (- \frac{6000 + 9659250000}{146250}) + 2 π 1

27 sin (- arccos (- \frac{6000 + 9659250000}{146250}) + 2 π 1) - 1.5 cos (- arccos (- \frac{6000 + 9659250000}{146250}) + 2 π 1) = 20

Refine

- 17.86089 \dots = 20

\Rightarrow False

Check the solution

arccos (\frac{9659250000 - 6000}{146250}) + 2 πn :

True

arccos (\frac{9659250000 - 6000}{146250}) + 2 πn

Plug in

n = 1

arccos (\frac{9659250000 - 6000}{146250}) + 2 π 1

For

27 sin (x) - 1.5 cos (x) = 20

plug in

x = arccos (\frac{9659250000 - 6000}{146250}) + 2 π 1

27 sin (arccos (\frac{9659250000 - 6000}{146250}) + 2 π 1) - 1.5 cos (arccos (\frac{9659250000 - 6000}{146250}) + 2 π 1) = 20

Refine

20 = 20

\Rightarrow True

Check the solution

2 π - arccos (\frac{9659250000 - 6000}{146250}) + 2 πn :

False

2 π - arccos (\frac{9659250000 - 6000}{146250}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{9659250000 - 6000}{146250}) + 2 π 1

For

27 sin (x) - 1.5 cos (x) = 20

plug in

x = 2 π - arccos (\frac{9659250000 - 6000}{146250}) + 2 π 1

27 sin (2 π - arccos (\frac{9659250000 - 6000}{146250}) + 2 π 1) - 1.5 cos (2 π - arccos (\frac{9659250000 - 6000}{146250}) + 2 π 1) = 20

Refine

- 21.89295 \dots = 20

\Rightarrow False

x = arccos (- \frac{6000 + 9659250000}{146250}) + 2 πn, x = arccos (\frac{9659250000 - 6000}{146250}) + 2 πn

Show solutions in decimal form

x = 2.36461 \dots + 2 πn, x = 0.88797 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 20=27sin(x)-1.5cos(x) ?
The general solution for 20=27sin(x)-1.5cos(x) is x=2.36461…+2pin,x=0.88797…+2pin