What is the general solution for 6193cos(x)+2880cos(2x)=0 ?

The general solution for 6193cos(x)+2880cos(2x)=0 is x=1.21251…+2pin,x=2pi-1.21251…+2pin

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Popular Trigonometry >

6193cos(x)+2880cos(2x)=0

Solution

6193 cos (x) + 2880 cos (2 x) = 0

Solution

x = 1.21251 \dots + 2 πn, x = 2 π - 1.21251 \dots + 2 πn

Degrees

x = 69.47171 \dots^{\circ} + 36 0^{\circ} n, x = 290.52828 \dots^{\circ} + 36 0^{\circ} n

Solution steps

6193 cos (x) + 2880 cos (2 x) = 0

Rewrite using trig identities

2880 cos (2 x) + 6193 cos (x)

Use the Double Angle identity:

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

= 2880 (2 cos^{2} (x) - 1) + 6193 cos (x)

(- 1 + 2 cos^{2} (x)) \cdot 2880 + 6193 cos (x) = 0

Solve by substitution

(- 1 + 2 cos^{2} (x)) \cdot 2880 + 6193 cos (x) = 0

Let:

cos (x) = u

(- 1 + 2 u^{2}) \cdot 2880 + 6193 u = 0

(- 1 + 2 u^{2}) \cdot 2880 + 6193 u = 0 : u = \frac{- 6193 + 104708449}{11520}, u = \frac{- 6193 - 104708449}{11520}

(- 1 + 2 u^{2}) \cdot 2880 + 6193 u = 0

Expand

(- 1 + 2 u^{2}) \cdot 2880 + 6193 u : - 2880 + 5760 u^{2} + 6193 u

(- 1 + 2 u^{2}) \cdot 2880 + 6193 u

= 2880 (- 1 + 2 u^{2}) + 6193 u

Expand

2880 (- 1 + 2 u^{2}) : - 2880 + 5760 u^{2}

2880 (- 1 + 2 u^{2})

Apply the distributive law:

a (b + c) = ab + a c

a = 2880, b = - 1, c = 2 u^{2}

= 2880 (- 1) + 2880 \cdot 2 u^{2}

Apply minus-plus rules

+ (- a) = - a

= - 2880 \cdot 1 + 2880 \cdot 2 u^{2}

Simplify

- 2880 \cdot 1 + 2880 \cdot 2 u^{2} : - 2880 + 5760 u^{2}

- 2880 \cdot 1 + 2880 \cdot 2 u^{2}

Multiply the numbers:

2880 \cdot 1 = 2880

= - 2880 + 2880 \cdot 2 u^{2}

Multiply the numbers:

2880 \cdot 2 = 5760

= - 2880 + 5760 u^{2}

= - 2880 + 5760 u^{2}

= - 2880 + 5760 u^{2} + 6193 u

- 2880 + 5760 u^{2} + 6193 u = 0

Divide both sides by

5760

- \frac{2880}{5760} + \frac{5760 u ^{2}}{5760} + \frac{6193 u}{5760} = \frac{0}{5760}

Write in the standard form

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

u^{2} + \frac{6193 u}{5760} - \frac{1}{2} = 0

Solve with the quadratic formula

u^{2} + \frac{6193 u}{5760} - \frac{1}{2} = 0

Quadratic Equation Formula:

For

a = 1, b = \frac{6193}{5760}, c = - \frac{1}{2}

u_{1, 2} = \frac{- \frac{6193}{5760} \pm ( \frac{6193}{5760} ) ^{2} - 4 \cdot 1 \cdot ( - \frac{1}{2} )}{2 \cdot 1}

u_{1, 2} = \frac{- \frac{6193}{5760} \pm ( \frac{6193}{5760} ) ^{2} - 4 \cdot 1 \cdot ( - \frac{1}{2} )}{2 \cdot 1}

(\frac{6193}{5760})^{2} - 4 \cdot 1 \cdot (- \frac{1}{2}) = \frac{104708449}{5760}

(\frac{6193}{5760})^{2} - 4 \cdot 1 \cdot (- \frac{1}{2})

Apply rule

- (- a) = a

= (\frac{6193}{5760})^{2} + 4 \cdot 1 \cdot \frac{1}{2}

(\frac{6193}{5760})^{2} = \frac{619 3 ^{2}}{576 0 ^{2}}

(\frac{6193}{5760})^{2}

Apply exponent rule:

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

= \frac{619 3 ^{2}}{576 0 ^{2}}

4 \cdot 1 \cdot \frac{1}{2} = 2

4 \cdot 1 \cdot \frac{1}{2}

Multiply fractions:

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

= 1 \cdot \frac{1 \cdot 4}{2}

\frac{1 \cdot 4}{2} = 2

\frac{1 \cdot 4}{2}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

= 1 \cdot 2

Multiply the numbers:

1 \cdot 2 = 2

= 2

= \frac{619 3 ^{2}}{576 0 ^{2}} + 2

\frac{619 3 ^{2}}{576 0 ^{2}} = \frac{38353249}{33177600}

\frac{619 3 ^{2}}{576 0 ^{2}}

619 3^{2} = 38353249

= \frac{38353249}{576 0 ^{2}}

576 0^{2} = 33177600

= \frac{38353249}{33177600}

= \frac{38353249}{33177600} + 2

Join

\frac{38353249}{33177600} + 2 : \frac{104708449}{33177600}

\frac{38353249}{33177600} + 2

Convert element to fraction:

2 = \frac{2 \cdot 33177600}{33177600}

= \frac{2 \cdot 33177600}{33177600} + \frac{38353249}{33177600}

Since the denominators are equal, combine the fractions:

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

= \frac{2 \cdot 33177600 + 38353249}{33177600}

2 \cdot 33177600 + 38353249 = 104708449

2 \cdot 33177600 + 38353249

Multiply the numbers:

2 \cdot 33177600 = 66355200

= 66355200 + 38353249

Add the numbers:

66355200 + 38353249 = 104708449

= 104708449

= \frac{104708449}{33177600}

= \frac{104708449}{33177600}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{104708449}{33177600}

33177600 = 5760

33177600

Factor the number:

33177600 = 576 0^{2}

= 576 0^{2}

Apply radical rule:

576 0^{2} = 5760

= 5760

= \frac{104708449}{5760}

u_{1, 2} = \frac{- \frac{6193}{5760} \pm \frac{104708449}{5760}}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- \frac{6193}{5760} + \frac{104708449}{5760}}{2 \cdot 1}, u_{2} = \frac{- \frac{6193}{5760} - \frac{104708449}{5760}}{2 \cdot 1}

u = \frac{- \frac{6193}{5760} + \frac{104708449}{5760}}{2 \cdot 1} : \frac{- 6193 + 104708449}{11520}

\frac{- \frac{6193}{5760} + \frac{104708449}{5760}}{2 \cdot 1}

Combine the fractions

- \frac{6193}{5760} + \frac{104708449}{5760} : \frac{- 6193 + 104708449}{5760}

Apply rule

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

= \frac{- 6193 + 104708449}{5760}

= \frac{\frac{- 6193 + 104708449}{5760}}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{\frac{- 6193 + 104708449}{5760}}{2}

Apply the fraction rule:

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

= \frac{- 6193 + 104708449}{5760 \cdot 2}

Multiply the numbers:

5760 \cdot 2 = 11520

= \frac{- 6193 + 104708449}{11520}

u = \frac{- \frac{6193}{5760} - \frac{104708449}{5760}}{2 \cdot 1} : \frac{- 6193 - 104708449}{11520}

\frac{- \frac{6193}{5760} - \frac{104708449}{5760}}{2 \cdot 1}

Combine the fractions

- \frac{6193}{5760} - \frac{104708449}{5760} : \frac{- 6193 - 104708449}{5760}

Apply rule

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

= \frac{- 6193 - 104708449}{5760}

= \frac{\frac{- 6193 - 104708449}{5760}}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{\frac{- 6193 - 104708449}{5760}}{2}

Apply the fraction rule:

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

= \frac{- 6193 - 104708449}{5760 \cdot 2}

Multiply the numbers:

5760 \cdot 2 = 11520

= \frac{- 6193 - 104708449}{11520}

The solutions to the quadratic equation are:

u = \frac{- 6193 + 104708449}{11520}, u = \frac{- 6193 - 104708449}{11520}

Substitute back

u = cos (x)

cos (x) = \frac{- 6193 + 104708449}{11520}, cos (x) = \frac{- 6193 - 104708449}{11520}

cos (x) = \frac{- 6193 + 104708449}{11520}, cos (x) = \frac{- 6193 - 104708449}{11520}

cos (x) = \frac{- 6193 + 104708449}{11520} : x = arccos (\frac{- 6193 + 104708449}{11520}) + 2 πn, x = 2 π - arccos (\frac{- 6193 + 104708449}{11520}) + 2 πn

cos (x) = \frac{- 6193 + 104708449}{11520}

Apply trig inverse properties

cos (x) = \frac{- 6193 + 104708449}{11520}

General solutions for

cos (x) = \frac{- 6193 + 104708449}{11520}

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

x = arccos (\frac{- 6193 + 104708449}{11520}) + 2 πn, x = 2 π - arccos (\frac{- 6193 + 104708449}{11520}) + 2 πn

x = arccos (\frac{- 6193 + 104708449}{11520}) + 2 πn, x = 2 π - arccos (\frac{- 6193 + 104708449}{11520}) + 2 πn

cos (x) = \frac{- 6193 - 104708449}{11520} :

No Solution

cos (x) = \frac{- 6193 - 104708449}{11520}

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = arccos (\frac{- 6193 + 104708449}{11520}) + 2 πn, x = 2 π - arccos (\frac{- 6193 + 104708449}{11520}) + 2 πn

Show solutions in decimal form

x = 1.21251 \dots + 2 πn, x = 2 π - 1.21251 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 6193cos(x)+2880cos(2x)=0 ?
The general solution for 6193cos(x)+2880cos(2x)=0 is x=1.21251…+2pin,x=2pi-1.21251…+2pin