What is the general solution for 2cos(x)-7=4sec(x) ?

The general solution for 2cos(x)-7=4sec(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin

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2cos(x)-7=4sec(x)

Solution

2 cos (x) - 7 = 4 sec (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

2 cos (x) - 7 = 4 sec (x)

Subtract

4 sec (x)

from both sides

2 cos (x) - 7 - 4 sec (x) = 0

Rewrite using trig identities

- 7 + 2 cos (x) - 4 sec (x)

Use the basic trigonometric identity:

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

= - 7 + 2 \cdot \frac{1}{sec ( x )} - 4 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 )}

= - 7 + \frac{2}{sec ( x )} - 4 sec (x)

- 7 + \frac{2}{sec ( x )} - 4 sec (x) = 0

Solve by substitution

- 7 + \frac{2}{sec ( x )} - 4 sec (x) = 0

Let:

sec (x) = u

- 7 + \frac{2}{u} - 4 u = 0

- 7 + \frac{2}{u} - 4 u = 0 : u = - 2, u = \frac{1}{4}

- 7 + \frac{2}{u} - 4 u = 0

Multiply both sides by

u

- 7 + \frac{2}{u} - 4 u = 0

Multiply both sides by

u

- 7 u + \frac{2}{u} u - 4 uu = 0 \cdot u

Simplify

- 7 u + \frac{2}{u} u - 4 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

- 4 uu : - 4 u^{2}

- 4 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= - 4 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= - 4 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

- 7 u + 2 - 4 u^{2} = 0

- 7 u + 2 - 4 u^{2} = 0

- 7 u + 2 - 4 u^{2} = 0

Solve

- 7 u + 2 - 4 u^{2} = 0 : u = - 2, u = \frac{1}{4}

- 7 u + 2 - 4 u^{2} = 0

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 4, b = - 7, c = 2

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

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

(- 7)^{2} - 4 (- 4) \cdot 2 = 9

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

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

= 7^{2} + 4 \cdot 4 \cdot 2

Multiply the numbers:

4 \cdot 4 \cdot 2 = 32

= 7^{2} + 32

7^{2} = 49

= 49 + 32

Add the numbers:

49 + 32 = 81

= 81

Factor the number:

81 = 9^{2}

= 9^{2}

Apply radical rule:

9^{2} = 9

= 9

u_{1, 2} = \frac{- ( - 7 ) \pm 9}{2 ( - 4 )}

Separate the solutions

u_{1} = \frac{- ( - 7 ) + 9}{2 ( - 4 )}, u_{2} = \frac{- ( - 7 ) - 9}{2 ( - 4 )}

u = \frac{- ( - 7 ) + 9}{2 ( - 4 )} : - 2

\frac{- ( - 7 ) + 9}{2 ( - 4 )}

Remove parentheses:

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

= \frac{7 + 9}{- 2 \cdot 4}

Add the numbers:

7 + 9 = 16

= \frac{16}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{16}{- 8}

Apply the fraction rule:

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

= - \frac{16}{8}

Divide the numbers:

\frac{16}{8} = 2

= - 2

u = \frac{- ( - 7 ) - 9}{2 ( - 4 )} : \frac{1}{4}

\frac{- ( - 7 ) - 9}{2 ( - 4 )}

Remove parentheses:

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

= \frac{7 - 9}{- 2 \cdot 4}

Subtract the numbers:

7 - 9 = - 2

= \frac{- 2}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 2}{- 8}

Apply the fraction rule:

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

= \frac{2}{8}

Cancel the common factor:

2

= \frac{1}{4}

The solutions to the quadratic equation are:

u = - 2, u = \frac{1}{4}

u = - 2, u = \frac{1}{4}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 7 + \frac{2}{u} - 4 u

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = - 2, u = \frac{1}{4}

Substitute back

u = sec (x)

sec (x) = - 2, sec (x) = \frac{1}{4}

sec (x) = - 2, sec (x) = \frac{1}{4}

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

sec (x) = \frac{1}{4} :

No Solution

sec (x) = \frac{1}{4}

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

No Solution

Combine all the solutions

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

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Frequently Asked Questions (FAQ)

What is the general solution for 2cos(x)-7=4sec(x) ?
The general solution for 2cos(x)-7=4sec(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin