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6sec^2(x)-3cos(x)-10=sec(x)

Solution

6 sec^{2} (x) - 3 cos (x) - 10 = sec (x)

Solution

x = π + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

Degrees

x = 18 0^{\circ} + 36 0^{\circ} n, x = 48.18968 \dots^{\circ} + 36 0^{\circ} n, x = 311.81031 \dots^{\circ} + 36 0^{\circ} n

Solution steps

6 sec^{2} (x) - 3 cos (x) - 10 = sec (x)

Subtract

sec (x)

from both sides

6 sec^{2} (x) - 3 cos (x) - 10 - sec (x) = 0

Rewrite using trig identities

- 10 - sec (x) - 3 cos (x) + 6 sec^{2} (x)

Use the basic trigonometric identity:

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

= - 10 - sec (x) - 3 \cdot \frac{1}{sec ( x )} + 6 sec^{2} (x)

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 3 = 3

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

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

- 10 - \frac{3}{sec ( x )} - sec (x) + 6 sec^{2} (x) = 0

Solve by substitution

- 10 - \frac{3}{sec ( x )} - sec (x) + 6 sec^{2} (x) = 0

Let:

sec (x) = u

- 10 - \frac{3}{u} - u + 6 u^{2} = 0

- 10 - \frac{3}{u} - u + 6 u^{2} = 0 : u = - 1, u = - \frac{1}{3}, u = \frac{3}{2}

- 10 - \frac{3}{u} - u + 6 u^{2} = 0

Multiply both sides by

u

- 10 - \frac{3}{u} - u + 6 u^{2} = 0

Multiply both sides by

u

- 10 u - \frac{3}{u} u - uu + 6 u^{2} u = 0 \cdot u

Simplify

- 10 u - \frac{3}{u} u - uu + 6 u^{2} u = 0 \cdot u

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

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

6 u^{2} u : 6 u^{3}

6 u^{2} u

Apply exponent rule:

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

u^{2} u = u^{2 + 1}

= 6 u^{2 + 1}

Add the numbers:

2 + 1 = 3

= 6 u^{3}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

- 10 u - 3 - u^{2} + 6 u^{3} = 0

- 10 u - 3 - u^{2} + 6 u^{3} = 0

- 10 u - 3 - u^{2} + 6 u^{3} = 0

Solve

- 10 u - 3 - u^{2} + 6 u^{3} = 0 : u = - 1, u = - \frac{1}{3}, u = \frac{3}{2}

- 10 u - 3 - u^{2} + 6 u^{3} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

6 u^{3} - u^{2} - 10 u - 3 = 0

Factor

6 u^{3} - u^{2} - 10 u - 3 : (u + 1) (3 u + 1) (2 u - 3)

6 u^{3} - u^{2} - 10 u - 3

Use the rational root theorem

a_{0} = 3, a_{n} = 6

The dividers of

a_{0} : 1, 3,

The dividers of

a_{n} : 1, 2, 3, 6

Therefore, check the following rational numbers:

\pm \frac{1 , 3}{1 , 2 , 3 , 6}

- \frac{1}{1}

is a root of the expression, so factor out

u + 1

= (u + 1) \frac{6 u ^{3} - u ^{2} - 10 u - 3}{u + 1}

\frac{6 u ^{3} - u ^{2} - 10 u - 3}{u + 1} = 6 u^{2} - 7 u - 3

\frac{6 u ^{3} - u ^{2} - 10 u - 3}{u + 1}

Divide

\frac{6 u ^{3} - u ^{2} - 10 u - 3}{u + 1} : \frac{6 u ^{3} - u ^{2} - 10 u - 3}{u + 1} = 6 u^{2} + \frac{- 7 u ^{2} - 10 u - 3}{u + 1}

Divide the leading coefficients of the numerator

6 u^{3} - u^{2} - 10 u - 3

and the divisor

u + 1 : \frac{6 u ^{3}}{u} = 6 u^{2}

Quotient = 6 u^{2}

Multiply

u + 1

6 u^{2} : 6 u^{3} + 6 u^{2}

Subtract

6 u^{3} + 6 u^{2}

from

6 u^{3} - u^{2} - 10 u - 3

to get new remainder

Remainder = - 7 u^{2} - 10 u - 3

Therefore

\frac{6 u ^{3} - u ^{2} - 10 u - 3}{u + 1} = 6 u^{2} + \frac{- 7 u ^{2} - 10 u - 3}{u + 1}

= 6 u^{2} + \frac{- 7 u ^{2} - 10 u - 3}{u + 1}

Divide

\frac{- 7 u ^{2} - 10 u - 3}{u + 1} : \frac{- 7 u ^{2} - 10 u - 3}{u + 1} = - 7 u + \frac{- 3 u - 3}{u + 1}

Divide the leading coefficients of the numerator

- 7 u^{2} - 10 u - 3

and the divisor

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

Quotient = - 7 u

Multiply

u + 1

- 7 u : - 7 u^{2} - 7 u

Subtract

- 7 u^{2} - 7 u

from

- 7 u^{2} - 10 u - 3

to get new remainder

Remainder = - 3 u - 3

Therefore

\frac{- 7 u ^{2} - 10 u - 3}{u + 1} = - 7 u + \frac{- 3 u - 3}{u + 1}

= 6 u^{2} - 7 u + \frac{- 3 u - 3}{u + 1}

Divide

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

Divide the leading coefficients of the numerator

- 3 u - 3

and the divisor

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

Quotient = - 3

Multiply

u + 1

- 3 : - 3 u - 3

Subtract

- 3 u - 3

from

- 3 u - 3

to get new remainder

Remainder = 0

Therefore

\frac{- 3 u - 3}{u + 1} = - 3

= 6 u^{2} - 7 u - 3

= 6 u^{2} - 7 u - 3

Factor

6 u^{2} - 7 u - 3 : (3 u + 1) (2 u - 3)

6 u^{2} - 7 u - 3

Break the expression into groups

6 u^{2} - 7 u - 3

Definition

Factors of

18 : 1, 2, 3, 6, 9, 18

18

Divisors (Factors)

Find the Prime factors of

18 : 2, 3, 3

18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3 \cdot 3

Multiply the prime factors of

18 : 6, 9

2 \cdot 3 = 6

3 \cdot 3 = 9

6, 9

6, 9

Add the prime factors:

2, 3

Add 1 and the number

18

itself

1, 18

The factors of

18

1, 2, 3, 6, 9, 18

Negative factors of

18 : - 1, - 2, - 3, - 6, - 9, - 18

Multiply the factors by

- 1

to get the negative factors

- 1, - 2, - 3, - 6, - 9, - 18

For every two factors such that

u * v = - 18,

check if

u + v = - 7

Check

u = 1, v = - 18 : u * v = - 18, u + v = - 17 \Rightarrow

FalseCheck

u = 2, v = - 9 : u * v = - 18, u + v = - 7 \Rightarrow

True

u = 2, v = - 9

Group into

(a x^{2} + ux) + (vx + c)

(6 u^{2} + 2 u) + (- 9 u - 3)

= (6 u^{2} + 2 u) + (- 9 u - 3)

Factor out

2 u

from

6 u^{2} + 2 u : 2 u (3 u + 1)

6 u^{2} + 2 u

Apply exponent rule:

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

u^{2} = uu

= 6 uu + 2 u

Rewrite

6

2 \cdot 3

= 2 \cdot 3 uu + 2 u

Factor out common term

2 u

= 2 u (3 u + 1)

Factor out

- 3

from

- 9 u - 3 : - 3 (3 u + 1)

- 9 u - 3

Rewrite

9

3 \cdot 3

= - 3 \cdot 3 u - 3

Factor out common term

- 3

= - 3 (3 u + 1)

= 2 u (3 u + 1) - 3 (3 u + 1)

Factor out common term

3 u + 1

= (3 u + 1) (2 u - 3)

= (u + 1) (3 u + 1) (2 u - 3)

(u + 1) (3 u + 1) (2 u - 3) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u + 1 = 0 or 3 u + 1 = 0 or 2 u - 3 = 0

Solve

u + 1 = 0 : u = - 1

u + 1 = 0

Move

1

to the right side

u + 1 = 0

Subtract

1

from both sides

u + 1 - 1 = 0 - 1

Simplify

u = - 1

u = - 1

Solve

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

3 u + 1 = 0

Move

1

to the right side

3 u + 1 = 0

Subtract

1

from both sides

3 u + 1 - 1 = 0 - 1

Simplify

3 u = - 1

3 u = - 1

Divide both sides by

3

3 u = - 1

Divide both sides by

3

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

Simplify

u = - \frac{1}{3}

u = - \frac{1}{3}

Solve

2 u - 3 = 0 : u = \frac{3}{2}

2 u - 3 = 0

Move

3

to the right side

2 u - 3 = 0

Add

3

to both sides

2 u - 3 + 3 = 0 + 3

Simplify

2 u = 3

2 u = 3

Divide both sides by

2

2 u = 3

Divide both sides by

2

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

Simplify

u = \frac{3}{2}

u = \frac{3}{2}

The solutions are

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

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

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 10 - \frac{3}{u} - u + 6 u^{2}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

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

Substitute back

u = sec (x)

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

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

sec (x) = - 1 : x = π + 2 πn

sec (x) = - 1

General solutions for

sec (x) = - 1

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 = π + 2 πn

x = π + 2 πn

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

sec (x) = \frac{3}{2}

Apply trig inverse properties

sec (x) = \frac{3}{2}

General solutions for

sec (x) = \frac{3}{2}

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = 2 π - arcsec (a) + 2 πn

x = arcsec (\frac{3}{2}) + 2 πn, x = 2 π - arcsec (\frac{3}{2}) + 2 πn

x = arcsec (\frac{3}{2}) + 2 πn, x = 2 π - arcsec (\frac{3}{2}) + 2 πn

Combine all the solutions

x = π + 2 πn, x = arcsec (\frac{3}{2}) + 2 πn, x = 2 π - arcsec (\frac{3}{2}) + 2 πn

Show solutions in decimal form

x = π + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

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