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

The general solution for sec(x)+4cos(x)=5 is x=1.31811…+2pin,x=2pi-1.31811…+2pin,x=2pin

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sec(x)+4cos(x)=5

Solution

sec (x) + 4 cos (x) = 5

Solution

x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn, x = 2 πn

Degrees

x = 75.52248 \dots^{\circ} + 36 0^{\circ} n, x = 284.47751 \dots^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n

Solution steps

sec (x) + 4 cos (x) = 5

Subtract

5

from both sides

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

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

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

Solve by substitution

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

Let:

sec (x) = u

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

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

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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

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

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

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

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

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

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

Solve

- 5 u + 4 + u^{2} = 0 : u = 4, u = 1

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 1, b = - 5, c = 4

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

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

(- 5)^{2} - 4 \cdot 1 \cdot 4 = 3

(- 5)^{2} - 4 \cdot 1 \cdot 4

Apply exponent rule:

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

n

is even

(- 5)^{2} = 5^{2}

= 5^{2} - 4 \cdot 1 \cdot 4

Multiply the numbers:

4 \cdot 1 \cdot 4 = 16

= 5^{2} - 16

5^{2} = 25

= 25 - 16

Subtract the numbers:

25 - 16 = 9

= 9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

u_{1, 2} = \frac{- ( - 5 ) \pm 3}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 5 ) + 3}{2 \cdot 1}, u_{2} = \frac{- ( - 5 ) - 3}{2 \cdot 1}

u = \frac{- ( - 5 ) + 3}{2 \cdot 1} : 4

\frac{- ( - 5 ) + 3}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{5 + 3}{2 \cdot 1}

Add the numbers:

5 + 3 = 8

= \frac{8}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{8}{2}

Divide the numbers:

\frac{8}{2} = 4

= 4

u = \frac{- ( - 5 ) - 3}{2 \cdot 1} : 1

\frac{- ( - 5 ) - 3}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{5 - 3}{2 \cdot 1}

Subtract the numbers:

5 - 3 = 2

= \frac{2}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

u = 4, u = 1

u = 4, u = 1

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 4, u = 1

Substitute back

u = sec (x)

sec (x) = 4, sec (x) = 1

sec (x) = 4, sec (x) = 1

sec (x) = 4 : x = arcsec (4) + 2 πn, x = 2 π - arcsec (4) + 2 πn

sec (x) = 4

Apply trig inverse properties

sec (x) = 4

General solutions for

sec (x) = 4

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

x = arcsec (4) + 2 πn, x = 2 π - arcsec (4) + 2 πn

x = arcsec (4) + 2 πn, x = 2 π - arcsec (4) + 2 πn

sec (x) = 1 : x = 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 = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

Combine all the solutions

x = arcsec (4) + 2 πn, x = 2 π - arcsec (4) + 2 πn, x = 2 πn

Show solutions in decimal form

x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn, x = 2 πn

Popular Examples

-sqrt(3)sec(x)=2

- 3 sec (x) = 2

3cos(x)csc(x)=2sqrt(3)cos(x)

3 cos (x) csc (x) = 23 cos (x)

cos(2x)= 9/41

cos (2 x) = \frac{9}{41}

sec(5x)= 2/(sqrt(3))

sec (5 x) = \frac{2}{3}

cos(a)=-4/9 , pi/2 <= a<= pi

cos (a) = - \frac{4}{9}, \frac{π}{2} \leq a \leq π

Frequently Asked Questions (FAQ)

What is the general solution for sec(x)+4cos(x)=5 ?
The general solution for sec(x)+4cos(x)=5 is x=1.31811…+2pin,x=2pi-1.31811…+2pin,x=2pin