What is the general solution for 3sec(x)+5-cos(x)=cos(x) ?

The general solution for 3sec(x)+5-cos(x)=cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin

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3sec(x)+5-cos(x)=cos(x)

Solution

3 sec (x) + 5 - cos (x) = cos (x)

Solution

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Degrees

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

Solution steps

3 sec (x) + 5 - cos (x) = cos (x)

Subtract

cos (x)

from both sides

3 sec (x) + 5 - 2 cos (x) = 0

Rewrite using trig identities

5 - 2 cos (x) + 3 sec (x)

Use the basic trigonometric identity:

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

= 5 - 2 \cdot \frac{1}{sec ( x )} + 3 sec (x)

2 \cdot \frac{1}{sec ( x )} = \frac{2}{sec ( x )}

2 \cdot \frac{1}{sec ( x )}

Multiply fractions:

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

= \frac{1 \cdot 2}{sec ( x )}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{sec ( x )}

= 5 - \frac{2}{sec ( x )} + 3 sec (x)

5 - \frac{2}{sec ( x )} + 3 sec (x) = 0

Solve by substitution

5 - \frac{2}{sec ( x )} + 3 sec (x) = 0

Let:

sec (x) = u

5 - \frac{2}{u} + 3 u = 0

5 - \frac{2}{u} + 3 u = 0 : u = \frac{1}{3}, u = - 2

5 - \frac{2}{u} + 3 u = 0

Multiply both sides by

u

5 - \frac{2}{u} + 3 u = 0

Multiply both sides by

u

5 u - \frac{2}{u} u + 3 uu = 0 \cdot u

Simplify

5 u - \frac{2}{u} u + 3 uu = 0 \cdot u

Simplify

- \frac{2}{u} u : - 2

- \frac{2}{u} u

Multiply fractions:

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

= - \frac{2 u}{u}

Cancel the common factor:

u

= - 2

Simplify

3 uu : 3 u^{2}

3 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 3 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 3 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

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

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

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

Solve

5 u - 2 + 3 u^{2} = 0 : u = \frac{1}{3}, u = - 2

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

Write in the standard form

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

3 u^{2} + 5 u - 2 = 0

Solve with the quadratic formula

3 u^{2} + 5 u - 2 = 0

Quadratic Equation Formula:

For

a = 3, b = 5, c = - 2

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 3 ( - 2 )}{2 \cdot 3}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 3 ( - 2 )}{2 \cdot 3}

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

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

Apply rule

- (- a) = a

= 5^{2} + 4 \cdot 3 \cdot 2

Multiply the numbers:

4 \cdot 3 \cdot 2 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

Add the numbers:

25 + 24 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

u_{1, 2} = \frac{- 5 \pm 7}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- 5 + 7}{2 \cdot 3}, u_{2} = \frac{- 5 - 7}{2 \cdot 3}

u = \frac{- 5 + 7}{2 \cdot 3} : \frac{1}{3}

\frac{- 5 + 7}{2 \cdot 3}

Add/Subtract the numbers:

- 5 + 7 = 2

= \frac{2}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{2}{6}

Cancel the common factor:

2

= \frac{1}{3}

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

\frac{- 5 - 7}{2 \cdot 3}

Subtract the numbers:

- 5 - 7 = - 12

= \frac{- 12}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 12}{6}

Apply the fraction rule:

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

= - \frac{12}{6}

Divide the numbers:

\frac{12}{6} = 2

= - 2

The solutions to the quadratic equation are:

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

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

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

5 - \frac{2}{u} + 3 u

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

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

Substitute back

u = sec (x)

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

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

sec (x) = \frac{1}{3} :

No Solution

sec (x) = \frac{1}{3}

sec (x) \leq - 1 or sec (x) \geq 1

No Solution

sec (x) = - 2 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

sec (x) = - 2

General solutions for

sec (x) = - 2

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Combine all the solutions

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Popular Examples

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

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

sin(2x-20)=-cos(3x+50)

sin (2 x - 20) = - cos (3 x + 50)

(sin(60))/(174.36)=(sin(x))/(200)

\frac{sin ( 6 0 ^{\circ} )}{174.36} = \frac{sin ( x )}{200}

cos(x)= 8/13

cos (x) = \frac{8}{13}

cos^2(4x)-sin^2(4x)=0

cos^{2} (4 x) - sin^{2} (4 x) = 0

Frequently Asked Questions (FAQ)

What is the general solution for 3sec(x)+5-cos(x)=cos(x) ?
The general solution for 3sec(x)+5-cos(x)=cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin