What is the general solution for 4/(cos(x))-6=tan(x) ?

The general solution for 4/(cos(x))-6=tan(x) is x=2pi-0.68802…+2pin,x=1.01832…+2pin

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Popular Trigonometry >

4/(cos(x))-6=tan(x)

Solution

\frac{4}{cos ( x )} - 6 = tan (x)

Solution

x = 2 π - 0.68802 \dots + 2 πn, x = 1.01832 \dots + 2 πn

Degrees

x = 320.57910 \dots^{\circ} + 36 0^{\circ} n, x = 58.34553 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{4}{cos ( x )} - 6 = tan (x)

Subtract

tan (x)

from both sides

\frac{4}{cos ( x )} - 6 - tan (x) = 0

Simplify

\frac{4}{cos ( x )} - 6 - tan (x) : \frac{4 - 6 cos ( x ) - tan ( x ) cos ( x )}{cos ( x )}

\frac{4}{cos ( x )} - 6 - tan (x)

Convert element to fraction:

6 = \frac{6 cos ( x )}{cos ( x )}, tan (x) = \frac{tan ( x ) cos ( x )}{cos ( x )}

= \frac{4}{cos ( x )} - \frac{6 cos ( x )}{cos ( x )} - \frac{tan ( x ) cos ( x )}{cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{4 - 6 cos ( x ) - tan ( x ) cos ( x )}{cos ( x )}

\frac{4 - 6 cos ( x ) - tan ( x ) cos ( x )}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

4 - 6 cos (x) - tan (x) cos (x) = 0

Express with sin, cos

4 - 6 cos (x) - \frac{sin ( x )}{cos ( x )} cos (x) = 0

Simplify

4 - 6 cos (x) - \frac{sin ( x )}{cos ( x )} cos (x) : 4 - 6 cos (x) - sin (x)

4 - 6 cos (x) - \frac{sin ( x )}{cos ( x )} cos (x)

\frac{sin ( x )}{cos ( x )} cos (x) = sin (x)

\frac{sin ( x )}{cos ( x )} cos (x)

Multiply fractions:

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

= \frac{sin ( x ) cos ( x )}{cos ( x )}

Cancel the common factor:

cos (x)

= sin (x)

= 4 - 6 cos (x) - sin (x)

4 - 6 cos (x) - sin (x) = 0

Add

sin (x)

to both sides

4 - 6 cos (x) = sin (x)

Square both sides

(4 - 6 cos (x))^{2} = sin^{2} (x)

Subtract

sin^{2} (x)

from both sides

(4 - 6 cos (x))^{2} - sin^{2} (x) = 0

Rewrite using trig identities

(4 - 6 cos (x))^{2} - sin^{2} (x)

Use the Pythagorean identity:

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

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

= (4 - 6 cos (x))^{2} - (1 - cos^{2} (x))

Simplify

(4 - 6 cos (x))^{2} - (1 - cos^{2} (x)) : 37 cos^{2} (x) - 48 cos (x) + 15

(4 - 6 cos (x))^{2} - (1 - cos^{2} (x))

(4 - 6 cos (x))^{2} : 16 - 48 cos (x) + 36 cos^{2} (x)

Apply Perfect Square Formula:

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

a = 4, b = 6 cos (x)

= 4^{2} - 2 \cdot 4 \cdot 6 cos (x) + (6 cos (x))^{2}

Simplify

4^{2} - 2 \cdot 4 \cdot 6 cos (x) + (6 cos (x))^{2} : 16 - 48 cos (x) + 36 cos^{2} (x)

4^{2} - 2 \cdot 4 \cdot 6 cos (x) + (6 cos (x))^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

2 \cdot 4 \cdot 6 cos (x) = 48 cos (x)

2 \cdot 4 \cdot 6 cos (x)

Multiply the numbers:

2 \cdot 4 \cdot 6 = 48

= 48 cos (x)

(6 cos (x))^{2} = 36 cos^{2} (x)

(6 cos (x))^{2}

Apply exponent rule:

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

= 6^{2} cos^{2} (x)

6^{2} = 36

= 36 cos^{2} (x)

= 16 - 48 cos (x) + 36 cos^{2} (x)

= 16 - 48 cos (x) + 36 cos^{2} (x)

= 16 - 48 cos (x) + 36 cos^{2} (x) - (1 - cos^{2} (x))

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

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

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

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

= 16 - 48 cos (x) + 36 cos^{2} (x) - 1 + cos^{2} (x)

Simplify

16 - 48 cos (x) + 36 cos^{2} (x) - 1 + cos^{2} (x) : 37 cos^{2} (x) - 48 cos (x) + 15

16 - 48 cos (x) + 36 cos^{2} (x) - 1 + cos^{2} (x)

Group like terms

= - 48 cos (x) + 36 cos^{2} (x) + cos^{2} (x) + 16 - 1

Add similar elements:

36 cos^{2} (x) + cos^{2} (x) = 37 cos^{2} (x)

= - 48 cos (x) + 37 cos^{2} (x) + 16 - 1

Add/Subtract the numbers:

16 - 1 = 15

= 37 cos^{2} (x) - 48 cos (x) + 15

= 37 cos^{2} (x) - 48 cos (x) + 15

= 37 cos^{2} (x) - 48 cos (x) + 15

15 + 37 cos^{2} (x) - 48 cos (x) = 0

Solve by substitution

15 + 37 cos^{2} (x) - 48 cos (x) = 0

Let:

cos (x) = u

15 + 37 u^{2} - 48 u = 0

15 + 37 u^{2} - 48 u = 0 : u = \frac{24 + 21}{37}, u = \frac{24 - 21}{37}

15 + 37 u^{2} - 48 u = 0

Write in the standard form

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

37 u^{2} - 48 u + 15 = 0

Solve with the quadratic formula

37 u^{2} - 48 u + 15 = 0

Quadratic Equation Formula:

For

a = 37, b = - 48, c = 15

u_{1, 2} = \frac{- ( - 48 ) \pm ( - 48 ) ^{2} - 4 \cdot 37 \cdot 15}{2 \cdot 37}

u_{1, 2} = \frac{- ( - 48 ) \pm ( - 48 ) ^{2} - 4 \cdot 37 \cdot 15}{2 \cdot 37}

(- 48)^{2} - 4 \cdot 37 \cdot 15 = 221

(- 48)^{2} - 4 \cdot 37 \cdot 15

Apply exponent rule:

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

n

is even

(- 48)^{2} = 4 8^{2}

= 4 8^{2} - 4 \cdot 37 \cdot 15

Multiply the numbers:

4 \cdot 37 \cdot 15 = 2220

= 4 8^{2} - 2220

4 8^{2} = 2304

= 2304 - 2220

Subtract the numbers:

2304 - 2220 = 84

= 84

Prime factorization of

84 : 2^{2} \cdot 3 \cdot 7

84

84

divides by

284 = 42 \cdot 2

= 2 \cdot 42

42

divides by

242 = 21 \cdot 2

= 2 \cdot 2 \cdot 21

21

divides by

321 = 7 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 7

2, 3, 7

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 7

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

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

Apply radical rule:

= 2^{2} 3 \cdot 7

Apply radical rule:

2^{2} = 2

= 2 3 \cdot 7

Refine

= 221

u_{1, 2} = \frac{- ( - 48 ) \pm 2 21}{2 \cdot 37}

Separate the solutions

u_{1} = \frac{- ( - 48 ) + 2 21}{2 \cdot 37}, u_{2} = \frac{- ( - 48 ) - 2 21}{2 \cdot 37}

u = \frac{- ( - 48 ) + 2 21}{2 \cdot 37} : \frac{24 + 21}{37}

\frac{- ( - 48 ) + 2 21}{2 \cdot 37}

Apply rule

- (- a) = a

= \frac{48 + 2 21}{2 \cdot 37}

Multiply the numbers:

2 \cdot 37 = 74

= \frac{48 + 2 21}{74}

Factor

48 + 221 : 2 (24 + 21)

48 + 221

Rewrite as

= 2 \cdot 24 + 221

Factor out common term

2

= 2 (24 + 21)

= \frac{2 ( 24 + 21 )}{74}

Cancel the common factor:

2

= \frac{24 + 21}{37}

u = \frac{- ( - 48 ) - 2 21}{2 \cdot 37} : \frac{24 - 21}{37}

\frac{- ( - 48 ) - 2 21}{2 \cdot 37}

Apply rule

- (- a) = a

= \frac{48 - 2 21}{2 \cdot 37}

Multiply the numbers:

2 \cdot 37 = 74

= \frac{48 - 2 21}{74}

Factor

48 - 221 : 2 (24 - 21)

48 - 221

Rewrite as

= 2 \cdot 24 - 221

Factor out common term

2

= 2 (24 - 21)

= \frac{2 ( 24 - 21 )}{74}

Cancel the common factor:

2

= \frac{24 - 21}{37}

The solutions to the quadratic equation are:

u = \frac{24 + 21}{37}, u = \frac{24 - 21}{37}

Substitute back

u = cos (x)

cos (x) = \frac{24 + 21}{37}, cos (x) = \frac{24 - 21}{37}

cos (x) = \frac{24 + 21}{37}, cos (x) = \frac{24 - 21}{37}

cos (x) = \frac{24 + 21}{37} : x = arccos (\frac{24 + 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn

cos (x) = \frac{24 + 21}{37}

Apply trig inverse properties

cos (x) = \frac{24 + 21}{37}

General solutions for

cos (x) = \frac{24 + 21}{37}

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

x = arccos (\frac{24 + 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn

x = arccos (\frac{24 + 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn

cos (x) = \frac{24 - 21}{37} : x = arccos (\frac{24 - 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 - 21}{37}) + 2 πn

cos (x) = \frac{24 - 21}{37}

Apply trig inverse properties

cos (x) = \frac{24 - 21}{37}

General solutions for

cos (x) = \frac{24 - 21}{37}

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

x = arccos (\frac{24 - 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 - 21}{37}) + 2 πn

x = arccos (\frac{24 - 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 - 21}{37}) + 2 πn

Combine all the solutions

x = arccos (\frac{24 + 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn, x = arccos (\frac{24 - 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 - 21}{37}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

\frac{4}{cos ( x )} - 6 = tan (x)

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

Check the solution

arccos (\frac{24 + 21}{37}) + 2 πn :

False

arccos (\frac{24 + 21}{37}) + 2 πn

Plug in

n = 1

arccos (\frac{24 + 21}{37}) + 2 π 1

For

\frac{4}{cos ( x )} - 6 = tan (x)

plug in

x = arccos (\frac{24 + 21}{37}) + 2 π 1

\frac{4}{cos ( arccos ( \frac{24 + 21}{37} ) + 2 π 1 )} - 6 = tan (arccos (\frac{24 + 21}{37}) + 2 π 1)

Refine

- 0.82202 \dots = 0.82202 \dots

\Rightarrow False

Check the solution

2 π - arccos (\frac{24 + 21}{37}) + 2 πn :

True

2 π - arccos (\frac{24 + 21}{37}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{24 + 21}{37}) + 2 π 1

For

\frac{4}{cos ( x )} - 6 = tan (x)

plug in

x = 2 π - arccos (\frac{24 + 21}{37}) + 2 π 1

\frac{4}{cos ( 2 π - arccos ( \frac{24 + 21}{37} ) + 2 π 1 )} - 6 = tan (2 π - arccos (\frac{24 + 21}{37}) + 2 π 1)

Refine

- 0.82202 \dots = - 0.82202 \dots

\Rightarrow True

Check the solution

arccos (\frac{24 - 21}{37}) + 2 πn :

True

arccos (\frac{24 - 21}{37}) + 2 πn

Plug in

n = 1

arccos (\frac{24 - 21}{37}) + 2 π 1

For

\frac{4}{cos ( x )} - 6 = tan (x)

plug in

x = arccos (\frac{24 - 21}{37}) + 2 π 1

\frac{4}{cos ( arccos ( \frac{24 - 21}{37} ) + 2 π 1 )} - 6 = tan (arccos (\frac{24 - 21}{37}) + 2 π 1)

Refine

1.62202 \dots = 1.62202 \dots

\Rightarrow True

Check the solution

2 π - arccos (\frac{24 - 21}{37}) + 2 πn :

False

2 π - arccos (\frac{24 - 21}{37}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{24 - 21}{37}) + 2 π 1

For

\frac{4}{cos ( x )} - 6 = tan (x)

plug in

x = 2 π - arccos (\frac{24 - 21}{37}) + 2 π 1

\frac{4}{cos ( 2 π - arccos ( \frac{24 - 21}{37} ) + 2 π 1 )} - 6 = tan (2 π - arccos (\frac{24 - 21}{37}) + 2 π 1)

Refine

1.62202 \dots = - 1.62202 \dots

\Rightarrow False

x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn, x = arccos (\frac{24 - 21}{37}) + 2 πn

Show solutions in decimal form

x = 2 π - 0.68802 \dots + 2 πn, x = 1.01832 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 4/(cos(x))-6=tan(x) ?
The general solution for 4/(cos(x))-6=tan(x) is x=2pi-0.68802…+2pin,x=1.01832…+2pin