What is the general solution for -11cos(x)22-11sin(x)8=-250 ?

The general solution for -11cos(x)22-11sin(x)8=-250 is x=0.10677…+2pin,x=0.59076…+2pin

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Popular Trigonometry >

-11cos(x)22-11sin(x)8=-250

Solution

- 11 cos (x) 22 - 11 sin (x) 8 = - 250

Solution

x = 0.10677 \dots + 2 πn, x = 0.59076 \dots + 2 πn

Degrees

x = 6.11761 \dots^{\circ} + 36 0^{\circ} n, x = 33.84860 \dots^{\circ} + 36 0^{\circ} n

Solution steps

- 11 cos (x) \cdot 22 - 11 sin (x) \cdot 8 = - 250

Add

11 sin (x) 8

to both sides

- 242 cos (x) = - 250 + 88 sin (x)

Square both sides

(- 242 cos (x))^{2} = (- 250 + 88 sin (x))^{2}

Subtract

(- 250 + 88 sin (x))^{2}

from both sides

58564 cos^{2} (x) - 62500 + 44000 sin (x) - 7744 sin^{2} (x) = 0

Rewrite using trig identities

- 62500 + 44000 sin (x) + 58564 cos^{2} (x) - 7744 sin^{2} (x)

Use the Pythagorean identity:

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

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

= - 62500 + 44000 sin (x) + 58564 (1 - sin^{2} (x)) - 7744 sin^{2} (x)

Simplify

- 62500 + 44000 sin (x) + 58564 (1 - sin^{2} (x)) - 7744 sin^{2} (x) : 44000 sin (x) - 66308 sin^{2} (x) - 3936

- 62500 + 44000 sin (x) + 58564 (1 - sin^{2} (x)) - 7744 sin^{2} (x)

Expand

58564 (1 - sin^{2} (x)) : 58564 - 58564 sin^{2} (x)

58564 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 58564, b = 1, c = sin^{2} (x)

= 58564 \cdot 1 - 58564 sin^{2} (x)

Multiply the numbers:

58564 \cdot 1 = 58564

= 58564 - 58564 sin^{2} (x)

= - 62500 + 44000 sin (x) + 58564 - 58564 sin^{2} (x) - 7744 sin^{2} (x)

Simplify

- 62500 + 44000 sin (x) + 58564 - 58564 sin^{2} (x) - 7744 sin^{2} (x) : 44000 sin (x) - 66308 sin^{2} (x) - 3936

- 62500 + 44000 sin (x) + 58564 - 58564 sin^{2} (x) - 7744 sin^{2} (x)

Add similar elements:

- 58564 sin^{2} (x) - 7744 sin^{2} (x) = - 66308 sin^{2} (x)

= - 62500 + 44000 sin (x) + 58564 - 66308 sin^{2} (x)

Group like terms

= 44000 sin (x) - 66308 sin^{2} (x) - 62500 + 58564

Add/Subtract the numbers:

- 62500 + 58564 = - 3936

= 44000 sin (x) - 66308 sin^{2} (x) - 3936

= 44000 sin (x) - 66308 sin^{2} (x) - 3936

= 44000 sin (x) - 66308 sin^{2} (x) - 3936

- 3936 + 44000 sin (x) - 66308 sin^{2} (x) = 0

Solve by substitution

- 3936 + 44000 sin (x) - 66308 sin^{2} (x) = 0

Let:

sin (x) = u

- 3936 + 44000 u - 66308 u^{2} = 0

- 3936 + 44000 u - 66308 u^{2} = 0 : u = \frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}, u = \frac{4400 0 ^{2} - 1043953152 + 44000}{132616}

- 3936 + 44000 u - 66308 u^{2} = 0

Write in the standard form

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

- 66308 u^{2} + 44000 u - 3936 = 0

Solve with the quadratic formula

- 66308 u^{2} + 44000 u - 3936 = 0

Quadratic Equation Formula:

For

a = - 66308, b = 44000, c = - 3936

u_{1, 2} = \frac{- 44000 \pm 4400 0 ^{2} - 4 ( - 66308 ) ( - 3936 )}{2 ( - 66308 )}

u_{1, 2} = \frac{- 44000 \pm 4400 0 ^{2} - 4 ( - 66308 ) ( - 3936 )}{2 ( - 66308 )}

4400 0^{2} - 4 (- 66308) (- 3936) = 4400 0^{2} - 1043953152

4400 0^{2} - 4 (- 66308) (- 3936)

Apply rule

- (- a) = a

= 4400 0^{2} - 4 \cdot 66308 \cdot 3936

Multiply the numbers:

4 \cdot 66308 \cdot 3936 = 1043953152

= 4400 0^{2} - 1043953152

u_{1, 2} = \frac{- 44000 \pm 4400 0 ^{2} - 1043953152}{2 ( - 66308 )}

Separate the solutions

u_{1} = \frac{- 44000 + 4400 0 ^{2} - 1043953152}{2 ( - 66308 )}, u_{2} = \frac{- 44000 - 4400 0 ^{2} - 1043953152}{2 ( - 66308 )}

u = \frac{- 44000 + 4400 0 ^{2} - 1043953152}{2 ( - 66308 )} : \frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}

\frac{- 44000 + 4400 0 ^{2} - 1043953152}{2 ( - 66308 )}

Remove parentheses:

(- a) = - a

= \frac{- 44000 + 4400 0 ^{2} - 1043953152}{- 2 \cdot 66308}

Multiply the numbers:

2 \cdot 66308 = 132616

= \frac{- 44000 + 4400 0 ^{2} - 1043953152}{- 132616}

Apply the fraction rule:

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

- 44000 + 4400 0^{2} - 1043953152 = - (- 4400 0^{2} - 1043953152 + 44000)

= \frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}

u = \frac{- 44000 - 4400 0 ^{2} - 1043953152}{2 ( - 66308 )} : \frac{4400 0 ^{2} - 1043953152 + 44000}{132616}

\frac{- 44000 - 4400 0 ^{2} - 1043953152}{2 ( - 66308 )}

Remove parentheses:

(- a) = - a

= \frac{- 44000 - 4400 0 ^{2} - 1043953152}{- 2 \cdot 66308}

Multiply the numbers:

2 \cdot 66308 = 132616

= \frac{- 44000 - 4400 0 ^{2} - 1043953152}{- 132616}

Apply the fraction rule:

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

- 44000 - 4400 0^{2} - 1043953152 = - (4400 0^{2} - 1043953152 + 44000)

= \frac{4400 0 ^{2} - 1043953152 + 44000}{132616}

The solutions to the quadratic equation are:

u = \frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}, u = \frac{4400 0 ^{2} - 1043953152 + 44000}{132616}

Substitute back

u = sin (x)

sin (x) = \frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}, sin (x) = \frac{4400 0 ^{2} - 1043953152 + 44000}{132616}

sin (x) = \frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}, sin (x) = \frac{4400 0 ^{2} - 1043953152 + 44000}{132616}

sin (x) = \frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616} : x = arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn, x = π - arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn

sin (x) = \frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}

Apply trig inverse properties

sin (x) = \frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}

General solutions for

sin (x) = \frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}

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

x = arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn, x = π - arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn

x = arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn, x = π - arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn

sin (x) = \frac{4400 0 ^{2} - 1043953152 + 44000}{132616} : x = arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn, x = π - arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn

sin (x) = \frac{4400 0 ^{2} - 1043953152 + 44000}{132616}

Apply trig inverse properties

sin (x) = \frac{4400 0 ^{2} - 1043953152 + 44000}{132616}

General solutions for

sin (x) = \frac{4400 0 ^{2} - 1043953152 + 44000}{132616}

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

x = arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn, x = π - arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn

x = arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn, x = π - arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn

Combine all the solutions

x = arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn, x = π - arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn, x = arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn, x = π - arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

- 11 cos (x) 22 - 11 sin (x) 8 = - 250

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

Check the solution

arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn :

True

arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn

Plug in

n = 1

arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 π 1

For

- 11 cos (x) 22 - 11 sin (x) 8 = - 250

plug in

x = arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 π 1

- 11 cos (arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 π 1) \cdot 22 - 11 sin (arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 π 1) \cdot 8 = - 250

Refine

- 250 = - 250

\Rightarrow True

Check the solution

π - arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn :

False

π - arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 π 1

For

- 11 cos (x) 22 - 11 sin (x) 8 = - 250

plug in

x = π - arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 π 1

- 11 cos (π - arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 π 1) \cdot 22 - 11 sin (π - arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 π 1) \cdot 8 = - 250

Refine

231.24373 \dots = - 250

\Rightarrow False

Check the solution

arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn :

True

arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn

Plug in

n = 1

arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 π 1

For

- 11 cos (x) 22 - 11 sin (x) 8 = - 250

plug in

x = arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 π 1

- 11 cos (arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 π 1) \cdot 22 - 11 sin (arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 π 1) \cdot 8 = - 250

Refine

- 250 = - 250

\Rightarrow True

Check the solution

π - arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn :

False

π - arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 π 1

For

- 11 cos (x) 22 - 11 sin (x) 8 = - 250

plug in

x = π - arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 π 1

- 11 cos (π - arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 π 1) \cdot 22 - 11 sin (π - arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 π 1) \cdot 8 = - 250

Refine

151.96794 \dots = - 250

\Rightarrow False

x = arcsin (\frac{- 4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn, x = arcsin (\frac{4400 0 ^{2} - 1043953152 + 44000}{132616}) + 2 πn

Show solutions in decimal form

x = 0.10677 \dots + 2 πn, x = 0.59076 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for -11cos(x)22-11sin(x)8=-250 ?
The general solution for -11cos(x)22-11sin(x)8=-250 is x=0.10677…+2pin,x=0.59076…+2pin