What is the general solution for 5sec(x)-4cos(x)=8 ?

The general solution for 5sec(x)-4cos(x)=8 is x= pi/3+2pin,x=(5pi)/3+2pin

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5sec(x)-4cos(x)=8

Solution

5 sec (x) - 4 cos (x) = 8

Solution

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

5 sec (x) - 4 cos (x) = 8

Subtract

8

from both sides

5 sec (x) - 4 cos (x) - 8 = 0

Rewrite using trig identities

- 8 - 4 cos (x) + 5 sec (x)

Use the basic trigonometric identity:

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

= - 8 - 4 \cdot \frac{1}{sec ( x )} + 5 sec (x)

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 4 = 4

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

= - 8 - \frac{4}{sec ( x )} + 5 sec (x)

- 8 - \frac{4}{sec ( x )} + 5 sec (x) = 0

Solve by substitution

- 8 - \frac{4}{sec ( x )} + 5 sec (x) = 0

Let:

sec (x) = u

- 8 - \frac{4}{u} + 5 u = 0

- 8 - \frac{4}{u} + 5 u = 0 : u = 2, u = - \frac{2}{5}

- 8 - \frac{4}{u} + 5 u = 0

Multiply both sides by

u

- 8 - \frac{4}{u} + 5 u = 0

Multiply both sides by

u

- 8 u - \frac{4}{u} u + 5 uu = 0 \cdot u

Simplify

- 8 u - \frac{4}{u} u + 5 uu = 0 \cdot u

Simplify

- \frac{4}{u} u : - 4

- \frac{4}{u} u

Multiply fractions:

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

= - \frac{4 u}{u}

Cancel the common factor:

u

= - 4

Simplify

5 uu : 5 u^{2}

5 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 5 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 5 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

- 8 u - 4 + 5 u^{2} = 0

- 8 u - 4 + 5 u^{2} = 0

- 8 u - 4 + 5 u^{2} = 0

Solve

- 8 u - 4 + 5 u^{2} = 0 : u = 2, u = - \frac{2}{5}

- 8 u - 4 + 5 u^{2} = 0

Write in the standard form

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

5 u^{2} - 8 u - 4 = 0

Solve with the quadratic formula

5 u^{2} - 8 u - 4 = 0

Quadratic Equation Formula:

For

a = 5, b = - 8, c = - 4

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 5 ( - 4 )}{2 \cdot 5}

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 5 ( - 4 )}{2 \cdot 5}

(- 8)^{2} - 4 \cdot 5 (- 4) = 12

(- 8)^{2} - 4 \cdot 5 (- 4)

Apply rule

- (- a) = a

= (- 8)^{2} + 4 \cdot 5 \cdot 4

Apply exponent rule:

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

n

is even

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

= 8^{2} + 4 \cdot 5 \cdot 4

Multiply the numbers:

4 \cdot 5 \cdot 4 = 80

= 8^{2} + 80

8^{2} = 64

= 64 + 80

Add the numbers:

64 + 80 = 144

= 144

Factor the number:

144 = 1 2^{2}

= 1 2^{2}

Apply radical rule:

1 2^{2} = 12

= 12

u_{1, 2} = \frac{- ( - 8 ) \pm 12}{2 \cdot 5}

Separate the solutions

u_{1} = \frac{- ( - 8 ) + 12}{2 \cdot 5}, u_{2} = \frac{- ( - 8 ) - 12}{2 \cdot 5}

u = \frac{- ( - 8 ) + 12}{2 \cdot 5} : 2

\frac{- ( - 8 ) + 12}{2 \cdot 5}

Apply rule

- (- a) = a

= \frac{8 + 12}{2 \cdot 5}

Add the numbers:

8 + 12 = 20

= \frac{20}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{20}{10}

Divide the numbers:

\frac{20}{10} = 2

= 2

u = \frac{- ( - 8 ) - 12}{2 \cdot 5} : - \frac{2}{5}

\frac{- ( - 8 ) - 12}{2 \cdot 5}

Apply rule

- (- a) = a

= \frac{8 - 12}{2 \cdot 5}

Subtract the numbers:

8 - 12 = - 4

= \frac{- 4}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{- 4}{10}

Apply the fraction rule:

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

= - \frac{4}{10}

Cancel the common factor:

2

= - \frac{2}{5}

The solutions to the quadratic equation are:

u = 2, u = - \frac{2}{5}

u = 2, u = - \frac{2}{5}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 8 - \frac{4}{u} + 5 u

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 2, u = - \frac{2}{5}

Substitute back

u = sec (x)

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

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

sec (x) = 2 : x = \frac{π}{3} + 2 πn, x = \frac{5 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{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

No Solution

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

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

No Solution

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 5sec(x)-4cos(x)=8 ?
The general solution for 5sec(x)-4cos(x)=8 is x= pi/3+2pin,x=(5pi)/3+2pin