What is the general solution for 6sec(2x)+3tan(2x)-9=0 ?

The general solution for 6sec(2x)+3tan(2x)-9=0 is x=(0.56432…)/2+pin,x=-(1.20782…)/2+pin

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Popular Trigonometry >

6sec(2x)+3tan(2x)-9=0

Solution

6 sec (2 x) + 3 tan (2 x) - 9 = 0

Solution

x = \frac{0.56432 \dots}{2} + πn, x = - \frac{1.20782 \dots}{2} + πn

Degrees

x = 16.16676 \dots^{\circ} + 18 0^{\circ} n, x = - 34.60171 \dots^{\circ} + 18 0^{\circ} n

Solution steps

6 sec (2 x) + 3 tan (2 x) - 9 = 0

Express with sin, cos

6 \cdot \frac{1}{cos ( 2 x )} + 3 \cdot \frac{sin ( 2 x )}{cos ( 2 x )} - 9 = 0

Simplify

6 \cdot \frac{1}{cos ( 2 x )} + 3 \cdot \frac{sin ( 2 x )}{cos ( 2 x )} - 9 : \frac{6 + 3 sin ( 2 x ) - 9 cos ( 2 x )}{cos ( 2 x )}

6 \cdot \frac{1}{cos ( 2 x )} + 3 \cdot \frac{sin ( 2 x )}{cos ( 2 x )} - 9

6 \cdot \frac{1}{cos ( 2 x )} = \frac{6}{cos ( 2 x )}

6 \cdot \frac{1}{cos ( 2 x )}

Multiply fractions:

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

= \frac{1 \cdot 6}{cos ( 2 x )}

Multiply the numbers:

1 \cdot 6 = 6

= \frac{6}{cos ( 2 x )}

3 \cdot \frac{sin ( 2 x )}{cos ( 2 x )} = \frac{3 sin ( 2 x )}{cos ( 2 x )}

3 \cdot \frac{sin ( 2 x )}{cos ( 2 x )}

Multiply fractions:

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

= \frac{sin ( 2 x ) \cdot 3}{cos ( 2 x )}

= \frac{6}{cos ( 2 x )} + \frac{3 sin ( 2 x )}{cos ( 2 x )} - 9

Combine the fractions

\frac{6}{cos ( 2 x )} + \frac{3 sin ( 2 x )}{cos ( 2 x )} : \frac{6 + 3 sin ( 2 x )}{cos ( 2 x )}

Apply rule

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

= \frac{6 + 3 sin ( 2 x )}{cos ( 2 x )}

= \frac{3 sin ( 2 x ) + 6}{cos ( 2 x )} - 9

Convert element to fraction:

9 = \frac{9 cos ( 2 x )}{cos ( 2 x )}

= \frac{6 + sin ( 2 x ) \cdot 3}{cos ( 2 x )} - \frac{9 cos ( 2 x )}{cos ( 2 x )}

Since the denominators are equal, combine the fractions:

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

= \frac{6 + sin ( 2 x ) \cdot 3 - 9 cos ( 2 x )}{cos ( 2 x )}

\frac{6 + 3 sin ( 2 x ) - 9 cos ( 2 x )}{cos ( 2 x )} = 0

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

6 + 3 sin (2 x) - 9 cos (2 x) = 0

Add

9 cos (2 x)

to both sides

6 + 3 sin (2 x) = 9 cos (2 x)

Square both sides

(6 + 3 sin (2 x))^{2} = (9 cos (2 x))^{2}

Subtract

(9 cos (2 x))^{2}

from both sides

(6 + 3 sin (2 x))^{2} - 81 cos^{2} (2 x) = 0

Rewrite using trig identities

(6 + 3 sin (2 x))^{2} - 81 cos^{2} (2 x)

Use the Pythagorean identity:

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

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

= (6 + 3 sin (2 x))^{2} - 81 (1 - sin^{2} (2 x))

Simplify

(6 + 3 sin (2 x))^{2} - 81 (1 - sin^{2} (2 x)) : 90 sin^{2} (2 x) + 36 sin (2 x) - 45

(6 + 3 sin (2 x))^{2} - 81 (1 - sin^{2} (2 x))

(6 + 3 sin (2 x))^{2} : 36 + 36 sin (2 x) + 9 sin^{2} (2 x)

Apply Perfect Square Formula:

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

a = 6, b = 3 sin (2 x)

= 6^{2} + 2 \cdot 6 \cdot 3 sin (2 x) + (3 sin (2 x))^{2}

Simplify

6^{2} + 2 \cdot 6 \cdot 3 sin (2 x) + (3 sin (2 x))^{2} : 36 + 36 sin (2 x) + 9 sin^{2} (2 x)

6^{2} + 2 \cdot 6 \cdot 3 sin (2 x) + (3 sin (2 x))^{2}

6^{2} = 36

6^{2}

6^{2} = 36

= 36

2 \cdot 6 \cdot 3 sin (2 x) = 36 sin (2 x)

2 \cdot 6 \cdot 3 sin (2 x)

Multiply the numbers:

2 \cdot 6 \cdot 3 = 36

= 36 sin (2 x)

(3 sin (2 x))^{2} = 9 sin^{2} (2 x)

(3 sin (2 x))^{2}

Apply exponent rule:

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

= 3^{2} sin^{2} (2 x)

3^{2} = 9

= 9 sin^{2} (2 x)

= 36 + 36 sin (2 x) + 9 sin^{2} (2 x)

= 36 + 36 sin (2 x) + 9 sin^{2} (2 x)

= 36 + 36 sin (2 x) + 9 sin^{2} (2 x) - 81 (1 - sin^{2} (2 x))

Expand

- 81 (1 - sin^{2} (2 x)) : - 81 + 81 sin^{2} (2 x)

- 81 (1 - sin^{2} (2 x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 81, b = 1, c = sin^{2} (2 x)

= - 81 \cdot 1 - (- 81) sin^{2} (2 x)

Apply minus-plus rules

- (- a) = a

= - 81 \cdot 1 + 81 sin^{2} (2 x)

Multiply the numbers:

81 \cdot 1 = 81

= - 81 + 81 sin^{2} (2 x)

= 36 + 36 sin (2 x) + 9 sin^{2} (2 x) - 81 + 81 sin^{2} (2 x)

Simplify

36 + 36 sin (2 x) + 9 sin^{2} (2 x) - 81 + 81 sin^{2} (2 x) : 90 sin^{2} (2 x) + 36 sin (2 x) - 45

36 + 36 sin (2 x) + 9 sin^{2} (2 x) - 81 + 81 sin^{2} (2 x)

Group like terms

= 36 sin (2 x) + 9 sin^{2} (2 x) + 81 sin^{2} (2 x) + 36 - 81

Add similar elements:

9 sin^{2} (2 x) + 81 sin^{2} (2 x) = 90 sin^{2} (2 x)

= 36 sin (2 x) + 90 sin^{2} (2 x) + 36 - 81

Add/Subtract the numbers:

36 - 81 = - 45

= 90 sin^{2} (2 x) + 36 sin (2 x) - 45

= 90 sin^{2} (2 x) + 36 sin (2 x) - 45

= 90 sin^{2} (2 x) + 36 sin (2 x) - 45

- 45 + 36 sin (2 x) + 90 sin^{2} (2 x) = 0

Solve by substitution

- 45 + 36 sin (2 x) + 90 sin^{2} (2 x) = 0

Let:

sin (2 x) = u

- 45 + 36 u + 90 u^{2} = 0

- 45 + 36 u + 90 u^{2} = 0 : u = \frac{- 2 + 3 6}{10}, u = - \frac{2 + 3 6}{10}

- 45 + 36 u + 90 u^{2} = 0

Write in the standard form

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

90 u^{2} + 36 u - 45 = 0

Solve with the quadratic formula

90 u^{2} + 36 u - 45 = 0

Quadratic Equation Formula:

For

a = 90, b = 36, c = - 45

u_{1, 2} = \frac{- 36 \pm 3 6 ^{2} - 4 \cdot 90 ( - 45 )}{2 \cdot 90}

u_{1, 2} = \frac{- 36 \pm 3 6 ^{2} - 4 \cdot 90 ( - 45 )}{2 \cdot 90}

3 6^{2} - 4 \cdot 90 (- 45) = 546

3 6^{2} - 4 \cdot 90 (- 45)

Apply rule

- (- a) = a

= 3 6^{2} + 4 \cdot 90 \cdot 45

Multiply the numbers:

4 \cdot 90 \cdot 45 = 16200

= 3 6^{2} + 16200

3 6^{2} = 1296

= 1296 + 16200

Add the numbers:

1296 + 16200 = 17496

= 17496

Prime factorization of

17496 : 2^{3} \cdot 3^{7}

17496

17496

divides by

217496 = 8748 \cdot 2

= 2 \cdot 8748

8748

divides by

28748 = 4374 \cdot 2

= 2 \cdot 2 \cdot 4374

4374

divides by

24374 = 2187 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2187

2187

divides by

32187 = 729 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 729

729

divides by

3729 = 243 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 243

243

divides by

3243 = 81 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 81

81

divides by

381 = 27 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 27

27

divides by

327 = 9 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3

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

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

Apply exponent rule:

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

= 3^{6} \cdot 2^{2} \cdot 2 \cdot 3

Apply radical rule:

= 2^{2} 3^{6} 2 \cdot 3

Apply radical rule:

2^{2} = 2

= 2 3^{6} 2 \cdot 3

Apply radical rule:

3^{6} = 3^{\frac{6}{2}} = 3^{3}

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

Refine

= 546

u_{1, 2} = \frac{- 36 \pm 54 6}{2 \cdot 90}

Separate the solutions

u_{1} = \frac{- 36 + 54 6}{2 \cdot 90}, u_{2} = \frac{- 36 - 54 6}{2 \cdot 90}

u = \frac{- 36 + 54 6}{2 \cdot 90} : \frac{- 2 + 3 6}{10}

\frac{- 36 + 54 6}{2 \cdot 90}

Multiply the numbers:

2 \cdot 90 = 180

= \frac{- 36 + 54 6}{180}

Factor

- 36 + 546 : 18 (- 2 + 36)

- 36 + 546

Rewrite as

= - 18 \cdot 2 + 18 \cdot 36

Factor out common term

18

= 18 (- 2 + 36)

= \frac{18 ( - 2 + 3 6 )}{180}

Cancel the common factor:

18

= \frac{- 2 + 3 6}{10}

u = \frac{- 36 - 54 6}{2 \cdot 90} : - \frac{2 + 3 6}{10}

\frac{- 36 - 54 6}{2 \cdot 90}

Multiply the numbers:

2 \cdot 90 = 180

= \frac{- 36 - 54 6}{180}

Factor

- 36 - 546 : - 18 (2 + 36)

- 36 - 546

Rewrite as

= - 18 \cdot 2 - 18 \cdot 36

Factor out common term

18

= - 18 (2 + 36)

= - \frac{18 ( 2 + 3 6 )}{180}

Cancel the common factor:

18

= - \frac{2 + 3 6}{10}

The solutions to the quadratic equation are:

u = \frac{- 2 + 3 6}{10}, u = - \frac{2 + 3 6}{10}

Substitute back

u = sin (2 x)

sin (2 x) = \frac{- 2 + 3 6}{10}, sin (2 x) = - \frac{2 + 3 6}{10}

sin (2 x) = \frac{- 2 + 3 6}{10}, sin (2 x) = - \frac{2 + 3 6}{10}

sin (2 x) = \frac{- 2 + 3 6}{10} : x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

sin (2 x) = \frac{- 2 + 3 6}{10}

Apply trig inverse properties

sin (2 x) = \frac{- 2 + 3 6}{10}

General solutions for

sin (2 x) = \frac{- 2 + 3 6}{10}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

2 x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, 2 x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

2 x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, 2 x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

Solve

2 x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn : x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

2 x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

Divide both sides by

2

2 x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

Solve

2 x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn : x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

2 x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

Divide both sides by

2

2 x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

sin (2 x) = - \frac{2 + 3 6}{10} : x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

sin (2 x) = - \frac{2 + 3 6}{10}

Apply trig inverse properties

sin (2 x) = - \frac{2 + 3 6}{10}

General solutions for

sin (2 x) = - \frac{2 + 3 6}{10}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

2 x = arcsin (- \frac{2 + 3 6}{10}) + 2 πn, 2 x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

2 x = arcsin (- \frac{2 + 3 6}{10}) + 2 πn, 2 x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

Solve

2 x = arcsin (- \frac{2 + 3 6}{10}) + 2 πn : x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

2 x = arcsin (- \frac{2 + 3 6}{10}) + 2 πn

Simplify

arcsin (- \frac{2 + 3 6}{10}) + 2 πn : - arcsin (\frac{2 + 3 6}{10}) + 2 πn

arcsin (- \frac{2 + 3 6}{10}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2 + 3 6}{10}) = - arcsin (\frac{2 + 3 6}{10})

= - arcsin (\frac{2 + 3 6}{10}) + 2 πn

2 x = - arcsin (\frac{2 + 3 6}{10}) + 2 πn

Divide both sides by

2

2 x = - arcsin (\frac{2 + 3 6}{10}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + \frac{2 πn}{2}

Simplify

x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

Solve

2 x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn : x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

2 x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

Divide both sides by

2

2 x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

Combine all the solutions

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn, x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

6 sec (2 x) + 3 tan (2 x) - 9 = 0

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

Check the solution

\frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn :

True

\frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

Plug in

n = 1

\frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1

For

6 sec (2 x) + 3 tan (2 x) - 9 = 0

plug in

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1

6 sec 2 \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1 + 3 tan 2 \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1 - 9 = 0

Refine

0 = 0

\Rightarrow True

Check the solution

\frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn :

False

\frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

Plug in

n = 1

\frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1

For

6 sec (2 x) + 3 tan (2 x) - 9 = 0

plug in

x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1

6 sec 2 \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1 + 3 tan 2 \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1 - 9 = 0

Refine

- 18 = 0

\Rightarrow False

Check the solution

- \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn :

True

- \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

Plug in

n = 1

- \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1

For

6 sec (2 x) + 3 tan (2 x) - 9 = 0

plug in

x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1

6 sec 2 - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1 + 3 tan 2 - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1 - 9 = 0

Refine

0 = 0

\Rightarrow True

Check the solution

\frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn :

False

\frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

Plug in

n = 1

\frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1

For

6 sec (2 x) + 3 tan (2 x) - 9 = 0

plug in

x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1

6 sec 2 \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1 + 3 tan 2 \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1 - 9 = 0

Refine

- 18 = 0

\Rightarrow False

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn, x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

Show solutions in decimal form

x = \frac{0.56432 \dots}{2} + πn, x = - \frac{1.20782 \dots}{2} + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 6sec(2x)+3tan(2x)-9=0 ?
The general solution for 6sec(2x)+3tan(2x)-9=0 is x=(0.56432…)/2+pin,x=-(1.20782…)/2+pin