What is the general solution for 400+290cos(x)+290sin(x)=0 ?

The general solution for 400+290cos(x)+290sin(x)=0 is x=-2.13356…+2pin,x=-2.57882…+2pin

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400+290cos(x)+290sin(x)=0

Solution

400 + 290 cos (x) + 290 sin (x) = 0

Solution

x = - 2.13356 \dots + 2 πn, x = - 2.57882 \dots + 2 πn

Degrees

x = - 122.24412 \dots^{\circ} + 36 0^{\circ} n, x = - 147.75587 \dots^{\circ} + 36 0^{\circ} n

Solution steps

400 + 290 cos (x) + 290 sin (x) = 0

Subtract

290 sin (x)

from both sides

400 + 290 cos (x) = - 290 sin (x)

Square both sides

(400 + 290 cos (x))^{2} = (- 290 sin (x))^{2}

Subtract

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

from both sides

(400 + 290 cos (x))^{2} - 84100 sin^{2} (x) = 0

Rewrite using trig identities

(400 + 290 cos (x))^{2} - 84100 sin^{2} (x)

Use the Pythagorean identity:

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

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

= (400 + 290 cos (x))^{2} - 84100 (1 - cos^{2} (x))

Simplify

(400 + 290 cos (x))^{2} - 84100 (1 - cos^{2} (x)) : 168200 cos^{2} (x) + 232000 cos (x) + 75900

(400 + 290 cos (x))^{2} - 84100 (1 - cos^{2} (x))

(400 + 290 cos (x))^{2} : 160000 + 232000 cos (x) + 84100 cos^{2} (x)

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = 400, b = 290 cos (x)

= 40 0^{2} + 2 \cdot 400 \cdot 290 cos (x) + (290 cos (x))^{2}

Simplify

40 0^{2} + 2 \cdot 400 \cdot 290 cos (x) + (290 cos (x))^{2} : 160000 + 232000 cos (x) + 84100 cos^{2} (x)

40 0^{2} + 2 \cdot 400 \cdot 290 cos (x) + (290 cos (x))^{2}

40 0^{2} = 160000

40 0^{2}

40 0^{2} = 160000

= 160000

2 \cdot 400 \cdot 290 cos (x) = 232000 cos (x)

2 \cdot 400 \cdot 290 cos (x)

Multiply the numbers:

2 \cdot 400 \cdot 290 = 232000

= 232000 cos (x)

(290 cos (x))^{2} = 84100 cos^{2} (x)

(290 cos (x))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 29 0^{2} cos^{2} (x)

29 0^{2} = 84100

= 84100 cos^{2} (x)

= 160000 + 232000 cos (x) + 84100 cos^{2} (x)

= 160000 + 232000 cos (x) + 84100 cos^{2} (x)

= 160000 + 232000 cos (x) + 84100 cos^{2} (x) - 84100 (1 - cos^{2} (x))

Expand

- 84100 (1 - cos^{2} (x)) : - 84100 + 84100 cos^{2} (x)

- 84100 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 84100, b = 1, c = cos^{2} (x)

= - 84100 \cdot 1 - (- 84100) cos^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 84100 \cdot 1 + 84100 cos^{2} (x)

Multiply the numbers:

84100 \cdot 1 = 84100

= - 84100 + 84100 cos^{2} (x)

= 160000 + 232000 cos (x) + 84100 cos^{2} (x) - 84100 + 84100 cos^{2} (x)

Simplify

160000 + 232000 cos (x) + 84100 cos^{2} (x) - 84100 + 84100 cos^{2} (x) : 168200 cos^{2} (x) + 232000 cos (x) + 75900

160000 + 232000 cos (x) + 84100 cos^{2} (x) - 84100 + 84100 cos^{2} (x)

Group like terms

= 232000 cos (x) + 84100 cos^{2} (x) + 84100 cos^{2} (x) + 160000 - 84100

Add similar elements:

84100 cos^{2} (x) + 84100 cos^{2} (x) = 168200 cos^{2} (x)

= 232000 cos (x) + 168200 cos^{2} (x) + 160000 - 84100

Add/Subtract the numbers:

160000 - 84100 = 75900

= 168200 cos^{2} (x) + 232000 cos (x) + 75900

= 168200 cos^{2} (x) + 232000 cos (x) + 75900

= 168200 cos^{2} (x) + 232000 cos (x) + 75900

75900 + 168200 cos^{2} (x) + 232000 cos (x) = 0

Solve by substitution

75900 + 168200 cos^{2} (x) + 232000 cos (x) = 0

Let:

cos (x) = u

75900 + 168200 u^{2} + 232000 u = 0

75900 + 168200 u^{2} + 232000 u = 0 : u = \frac{- 40 + 82}{58}, u = \frac{- 40 - 82}{58}

75900 + 168200 u^{2} + 232000 u = 0

Divide both sides by

168200

\frac{75900}{168200} + \frac{168200 u ^{2}}{168200} + \frac{232000 u}{168200} = \frac{0}{168200}

Write in the standard form

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

u^{2} + \frac{40 u}{29} + \frac{759}{1682} = 0

Solve with the quadratic formula

u^{2} + \frac{40 u}{29} + \frac{759}{1682} = 0

Quadratic Equation Formula:

For

a = 1, b = \frac{40}{29}, c = \frac{759}{1682}

u_{1, 2} = \frac{- \frac{40}{29} \pm ( \frac{40}{29} ) ^{2} - 4 \cdot 1 \cdot \frac{759}{1682}}{2 \cdot 1}

u_{1, 2} = \frac{- \frac{40}{29} \pm ( \frac{40}{29} ) ^{2} - 4 \cdot 1 \cdot \frac{759}{1682}}{2 \cdot 1}

(\frac{40}{29})^{2} - 4 \cdot 1 \cdot \frac{759}{1682} = \frac{82}{29}

(\frac{40}{29})^{2} - 4 \cdot 1 \cdot \frac{759}{1682}

(\frac{40}{29})^{2} = \frac{4 0 ^{2}}{2 9 ^{2}}

(\frac{40}{29})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{4 0 ^{2}}{2 9 ^{2}}

4 \cdot 1 \cdot \frac{759}{1682} = \frac{1518}{841}

4 \cdot 1 \cdot \frac{759}{1682}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= 1 \cdot \frac{759 \cdot 4}{1682}

\frac{759 \cdot 4}{1682} = \frac{1518}{841}

\frac{759 \cdot 4}{1682}

Multiply the numbers:

759 \cdot 4 = 3036

= \frac{3036}{1682}

Cancel the common factor:

2

= \frac{1518}{841}

= 1 \cdot \frac{1518}{841}

Multiply:

1 \cdot \frac{1518}{841} = \frac{1518}{841}

= \frac{1518}{841}

= \frac{4 0 ^{2}}{2 9 ^{2}} - \frac{1518}{841}

\frac{4 0 ^{2}}{2 9 ^{2}} = \frac{1600}{841}

\frac{4 0 ^{2}}{2 9 ^{2}}

4 0^{2} = 1600

= \frac{1600}{2 9 ^{2}}

2 9^{2} = 841

= \frac{1600}{841}

= \frac{1600}{841} - \frac{1518}{841}

Combine the fractions

\frac{1600}{841} - \frac{1518}{841} : \frac{82}{841}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1600 - 1518}{841}

Subtract the numbers:

1600 - 1518 = 82

= \frac{82}{841}

= \frac{82}{841}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{82}{841}

841 = 29

841

Factor the number:

841 = 2 9^{2}

= 2 9^{2}

Apply radical rule:

2 9^{2} = 29

= 29

= \frac{82}{29}

u_{1, 2} = \frac{- \frac{40}{29} \pm \frac{82}{29}}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- \frac{40}{29} + \frac{82}{29}}{2 \cdot 1}, u_{2} = \frac{- \frac{40}{29} - \frac{82}{29}}{2 \cdot 1}

u = \frac{- \frac{40}{29} + \frac{82}{29}}{2 \cdot 1} : \frac{- 40 + 82}{58}

\frac{- \frac{40}{29} + \frac{82}{29}}{2 \cdot 1}

Combine the fractions

- \frac{40}{29} + \frac{82}{29} : \frac{- 40 + 82}{29}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 40 + 82}{29}

= \frac{\frac{- 40 + 82}{29}}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{\frac{- 40 + 82}{29}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{- 40 + 82}{29 \cdot 2}

Multiply the numbers:

29 \cdot 2 = 58

= \frac{- 40 + 82}{58}

u = \frac{- \frac{40}{29} - \frac{82}{29}}{2 \cdot 1} : \frac{- 40 - 82}{58}

\frac{- \frac{40}{29} - \frac{82}{29}}{2 \cdot 1}

Combine the fractions

- \frac{40}{29} - \frac{82}{29} : \frac{- 40 - 82}{29}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 40 - 82}{29}

= \frac{\frac{- 40 - 82}{29}}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{\frac{- 40 - 82}{29}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{- 40 - 82}{29 \cdot 2}

Multiply the numbers:

29 \cdot 2 = 58

= \frac{- 40 - 82}{58}

The solutions to the quadratic equation are:

u = \frac{- 40 + 82}{58}, u = \frac{- 40 - 82}{58}

Substitute back

u = cos (x)

cos (x) = \frac{- 40 + 82}{58}, cos (x) = \frac{- 40 - 82}{58}

cos (x) = \frac{- 40 + 82}{58}, cos (x) = \frac{- 40 - 82}{58}

cos (x) = \frac{- 40 + 82}{58} : x = arccos (\frac{- 40 + 82}{58}) + 2 πn, x = - arccos (\frac{- 40 + 82}{58}) + 2 πn

cos (x) = \frac{- 40 + 82}{58}

Apply trig inverse properties

cos (x) = \frac{- 40 + 82}{58}

General solutions for

cos (x) = \frac{- 40 + 82}{58}

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

x = arccos (\frac{- 40 + 82}{58}) + 2 πn, x = - arccos (\frac{- 40 + 82}{58}) + 2 πn

x = arccos (\frac{- 40 + 82}{58}) + 2 πn, x = - arccos (\frac{- 40 + 82}{58}) + 2 πn

cos (x) = \frac{- 40 - 82}{58} : x = arccos (\frac{- 40 - 82}{58}) + 2 πn, x = - arccos (\frac{- 40 - 82}{58}) + 2 πn

cos (x) = \frac{- 40 - 82}{58}

Apply trig inverse properties

cos (x) = \frac{- 40 - 82}{58}

General solutions for

cos (x) = \frac{- 40 - 82}{58}

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

x = arccos (\frac{- 40 - 82}{58}) + 2 πn, x = - arccos (\frac{- 40 - 82}{58}) + 2 πn

x = arccos (\frac{- 40 - 82}{58}) + 2 πn, x = - arccos (\frac{- 40 - 82}{58}) + 2 πn

Combine all the solutions

x = arccos (\frac{- 40 + 82}{58}) + 2 πn, x = - arccos (\frac{- 40 + 82}{58}) + 2 πn, x = arccos (\frac{- 40 - 82}{58}) + 2 πn, x = - arccos (\frac{- 40 - 82}{58}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

400 + 290 cos (x) + 290 sin (x) = 0

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

Check the solution

arccos (\frac{- 40 + 82}{58}) + 2 πn :

False

arccos (\frac{- 40 + 82}{58}) + 2 πn

Plug in

n = 1

arccos (\frac{- 40 + 82}{58}) + 2 π 1

For

400 + 290 cos (x) + 290 sin (x) = 0

plug in

x = arccos (\frac{- 40 + 82}{58}) + 2 π 1

400 + 290 cos (arccos (\frac{- 40 + 82}{58}) + 2 π 1) + 290 sin (arccos (\frac{- 40 + 82}{58}) + 2 π 1) = 0

Refine

490.55385 \dots = 0

\Rightarrow False

Check the solution

- arccos (\frac{- 40 + 82}{58}) + 2 πn :

True

- arccos (\frac{- 40 + 82}{58}) + 2 πn

Plug in

n = 1

- arccos (\frac{- 40 + 82}{58}) + 2 π 1

For

400 + 290 cos (x) + 290 sin (x) = 0

plug in

x = - arccos (\frac{- 40 + 82}{58}) + 2 π 1

400 + 290 cos (- arccos (\frac{- 40 + 82}{58}) + 2 π 1) + 290 sin (- arccos (\frac{- 40 + 82}{58}) + 2 π 1) = 0

Refine

0 = 0

\Rightarrow True

Check the solution

arccos (\frac{- 40 - 82}{58}) + 2 πn :

False

arccos (\frac{- 40 - 82}{58}) + 2 πn

Plug in

n = 1

arccos (\frac{- 40 - 82}{58}) + 2 π 1

For

400 + 290 cos (x) + 290 sin (x) = 0

plug in

x = arccos (\frac{- 40 - 82}{58}) + 2 π 1

400 + 290 cos (arccos (\frac{- 40 - 82}{58}) + 2 π 1) + 290 sin (arccos (\frac{- 40 - 82}{58}) + 2 π 1) = 0

Refine

309.44614 \dots = 0

\Rightarrow False

Check the solution

- arccos (\frac{- 40 - 82}{58}) + 2 πn :

True

- arccos (\frac{- 40 - 82}{58}) + 2 πn

Plug in

n = 1

- arccos (\frac{- 40 - 82}{58}) + 2 π 1

For

400 + 290 cos (x) + 290 sin (x) = 0

plug in

x = - arccos (\frac{- 40 - 82}{58}) + 2 π 1

400 + 290 cos (- arccos (\frac{- 40 - 82}{58}) + 2 π 1) + 290 sin (- arccos (\frac{- 40 - 82}{58}) + 2 π 1) = 0

Refine

0 = 0

\Rightarrow True

x = - arccos (\frac{- 40 + 82}{58}) + 2 πn, x = - arccos (\frac{- 40 - 82}{58}) + 2 πn

Show solutions in decimal form

x = - 2.13356 \dots + 2 πn, x = - 2.57882 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 400+290cos(x)+290sin(x)=0 ?
The general solution for 400+290cos(x)+290sin(x)=0 is x=-2.13356…+2pin,x=-2.57882…+2pin