What is the general solution for 2cos(x)=7-3/(cos(x)) ?

The general solution for 2cos(x)=7-3/(cos(x)) is x= pi/3+2pin,x=(5pi)/3+2pin

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2cos(x)=7-3/(cos(x))

Solution

2 cos (x) = 7 - \frac{3}{cos ( x )}

Solution

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

2 cos (x) = 7 - \frac{3}{cos ( x )}

Solve by substitution

2 cos (x) = 7 - \frac{3}{cos ( x )}

Let:

cos (x) = u

2 u = 7 - \frac{3}{u}

2 u = 7 - \frac{3}{u} : u = 3, u = \frac{1}{2}

2 u = 7 - \frac{3}{u}

Multiply both sides by

u

2 u = 7 - \frac{3}{u}

Multiply both sides by

u

2 uu = 7 u - \frac{3}{u} u

Simplify

2 uu = 7 u - \frac{3}{u} u

Simplify

2 uu : 2 u^{2}

2 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 2 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2 u^{2}

Simplify

- \frac{3}{u} u : - 3

- \frac{3}{u} u

Multiply fractions:

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

= - \frac{3 u}{u}

Cancel the common factor:

u

= - 3

2 u^{2} = 7 u - 3

2 u^{2} = 7 u - 3

2 u^{2} = 7 u - 3

Solve

2 u^{2} = 7 u - 3 : u = 3, u = \frac{1}{2}

2 u^{2} = 7 u - 3

Move

3

to the left side

2 u^{2} = 7 u - 3

Add

3

to both sides

2 u^{2} + 3 = 7 u - 3 + 3

Simplify

2 u^{2} + 3 = 7 u

2 u^{2} + 3 = 7 u

Move

7 u

to the left side

2 u^{2} + 3 = 7 u

Subtract

7 u

from both sides

2 u^{2} + 3 - 7 u = 7 u - 7 u

Simplify

2 u^{2} + 3 - 7 u = 0

2 u^{2} + 3 - 7 u = 0

Write in the standard form

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

2 u^{2} - 7 u + 3 = 0

Solve with the quadratic formula

2 u^{2} - 7 u + 3 = 0

Quadratic Equation Formula:

For

a = 2, b = - 7, c = 3

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 2 \cdot 3}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 2 \cdot 3}{2 \cdot 2}

(- 7)^{2} - 4 \cdot 2 \cdot 3 = 5

(- 7)^{2} - 4 \cdot 2 \cdot 3

Apply exponent rule:

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

n

is even

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 2 \cdot 3

Multiply the numbers:

4 \cdot 2 \cdot 3 = 24

= 7^{2} - 24

7^{2} = 49

= 49 - 24

Subtract the numbers:

49 - 24 = 25

= 25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- ( - 7 ) \pm 5}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- ( - 7 ) + 5}{2 \cdot 2}, u_{2} = \frac{- ( - 7 ) - 5}{2 \cdot 2}

u = \frac{- ( - 7 ) + 5}{2 \cdot 2} : 3

\frac{- ( - 7 ) + 5}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{7 + 5}{2 \cdot 2}

Add the numbers:

7 + 5 = 12

= \frac{12}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{12}{4}

Divide the numbers:

\frac{12}{4} = 3

= 3

u = \frac{- ( - 7 ) - 5}{2 \cdot 2} : \frac{1}{2}

\frac{- ( - 7 ) - 5}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{7 - 5}{2 \cdot 2}

Subtract the numbers:

7 - 5 = 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

The solutions to the quadratic equation are:

u = 3, u = \frac{1}{2}

u = 3, u = \frac{1}{2}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

7 - \frac{3}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 3, u = \frac{1}{2}

Substitute back

u = cos (x)

cos (x) = 3, cos (x) = \frac{1}{2}

cos (x) = 3, cos (x) = \frac{1}{2}

cos (x) = 3 :

No Solution

cos (x) = 3

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

cos (x) = \frac{1}{2}

General solutions for

cos (x) = \frac{1}{2}

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 = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Graph

Popular Examples

-sin(x)+2=0

- sin (x) + 2 = 0

csc(x)=-9/7 ,tan(x)>0

csc (x) = - \frac{9}{7}, tan (x) > 0

sin(x)= 12/37

sin (x) = \frac{12}{37}

tan(θ)=(-2sqrt(3))/2

tan (θ) = \frac{- 2 3}{2}

sec(4x)-9=0

sec (4 x) - 9 = 0

Frequently Asked Questions (FAQ)

What is the general solution for 2cos(x)=7-3/(cos(x)) ?
The general solution for 2cos(x)=7-3/(cos(x)) is x= pi/3+2pin,x=(5pi)/3+2pin