What is the general solution for cos(x)=-3+4/(cos(x)) ?

The general solution for cos(x)=-3+4/(cos(x)) is x=2pin

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cos(x)=-3+4/(cos(x))

Solution

cos (x) = - 3 + \frac{4}{cos ( x )}

Solution

x = 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n

Solution steps

cos (x) = - 3 + \frac{4}{cos ( x )}

Solve by substitution

cos (x) = - 3 + \frac{4}{cos ( x )}

Let:

cos (x) = u

u = - 3 + \frac{4}{u}

u = - 3 + \frac{4}{u} : u = 1, u = - 4

u = - 3 + \frac{4}{u}

Multiply both sides by

u

u = - 3 + \frac{4}{u}

Multiply both sides by

u

uu = - 3 u + \frac{4}{u} u

Simplify

uu = - 3 u + \frac{4}{u} u

Simplify

uu : u^{2}

uu

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

Simplify

\frac{4}{u} u : 4

\frac{4}{u} u

Multiply fractions:

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

= \frac{4 u}{u}

Cancel the common factor:

u

= 4

u^{2} = - 3 u + 4

u^{2} = - 3 u + 4

u^{2} = - 3 u + 4

Solve

u^{2} = - 3 u + 4 : u = 1, u = - 4

u^{2} = - 3 u + 4

Move

4

to the left side

u^{2} = - 3 u + 4

Subtract

4

from both sides

u^{2} - 4 = - 3 u + 4 - 4

Simplify

u^{2} - 4 = - 3 u

u^{2} - 4 = - 3 u

Move

3 u

to the left side

u^{2} - 4 = - 3 u

Add

3 u

to both sides

u^{2} - 4 + 3 u = - 3 u + 3 u

Simplify

u^{2} - 4 + 3 u = 0

u^{2} - 4 + 3 u = 0

Write in the standard form

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

u^{2} + 3 u - 4 = 0

Solve with the quadratic formula

u^{2} + 3 u - 4 = 0

Quadratic Equation Formula:

For

a = 1, b = 3, c = - 4

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 1 \cdot ( - 4 )}{2 \cdot 1}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 1 \cdot ( - 4 )}{2 \cdot 1}

3^{2} - 4 \cdot 1 \cdot (- 4) = 5

3^{2} - 4 \cdot 1 \cdot (- 4)

Apply rule

- (- a) = a

= 3^{2} + 4 \cdot 1 \cdot 4

Multiply the numbers:

4 \cdot 1 \cdot 4 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

Add the numbers:

9 + 16 = 25

= 25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

5^{2} = 5

= 5

u_{1, 2} = \frac{- 3 \pm 5}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 3 + 5}{2 \cdot 1}, u_{2} = \frac{- 3 - 5}{2 \cdot 1}

u = \frac{- 3 + 5}{2 \cdot 1} : 1

\frac{- 3 + 5}{2 \cdot 1}

Add/Subtract the numbers:

- 3 + 5 = 2

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

u = \frac{- 3 - 5}{2 \cdot 1} : - 4

\frac{- 3 - 5}{2 \cdot 1}

Subtract the numbers:

- 3 - 5 = - 8

= \frac{- 8}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 8}{2}

Apply the fraction rule:

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

= - \frac{8}{2}

Divide the numbers:

\frac{8}{2} = 4

= - 4

The solutions to the quadratic equation are:

u = 1, u = - 4

u = 1, u = - 4

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 3 + \frac{4}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 1, u = - 4

Substitute back

u = cos (x)

cos (x) = 1, cos (x) = - 4

cos (x) = 1, cos (x) = - 4

cos (x) = 1 : x = 2 πn

cos (x) = 1

General solutions for

cos (x) = 1

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

cos (x) = - 4 :

No Solution

cos (x) = - 4

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = 2 πn

Popular Examples

sin(x)+4cos(x)+5=0

sin (x) + 4 cos (x) + 5 = 0

sec(x)=-5

sec (x) = - 5

sin(2t-pi/5)=-1/3

sin (2 t - \frac{π}{5}) = - \frac{1}{3}

2(sin(x))^2-cos(x)-1=0

2 (sin (x))^{2} - cos (x) - 1 = 0

arctan(3x)+arctan(x)= pi/4

arctan (3 x) + arctan (x) = \frac{π}{4}

Frequently Asked Questions (FAQ)

What is the general solution for cos(x)=-3+4/(cos(x)) ?
The general solution for cos(x)=-3+4/(cos(x)) is x=2pin