What is the general solution for 4*cos(x)+3*sec(x)=8 ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

4cos(x)+3sec(x)=8

Solution

4 \cdot cos (x) + 3 \cdot sec (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

4 cos (x) + 3 sec (x) = 8

Subtract

8

from both sides

4 cos (x) + 3 sec (x) - 8 = 0

Rewrite using trig identities

- 8 + 3 sec (x) + 4 cos (x)

Use the basic trigonometric identity:

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

= - 8 + 3 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 )}

= - 8 + 3 sec (x) + \frac{4}{sec ( x )}

- 8 + \frac{4}{sec ( x )} + 3 sec (x) = 0

Solve by substitution

- 8 + \frac{4}{sec ( x )} + 3 sec (x) = 0

Let:

sec (x) = u

- 8 + \frac{4}{u} + 3 u = 0

- 8 + \frac{4}{u} + 3 u = 0 : u = 2, u = \frac{2}{3}

- 8 + \frac{4}{u} + 3 u = 0

Multiply both sides by

u

- 8 + \frac{4}{u} + 3 u = 0

Multiply both sides by

u

- 8 u + \frac{4}{u} u + 3 uu = 0 \cdot u

Simplify

- 8 u + \frac{4}{u} u + 3 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

3 uu : 3 u^{2}

3 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 3 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 3 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

- 8 u + 4 + 3 u^{2} = 0

- 8 u + 4 + 3 u^{2} = 0

- 8 u + 4 + 3 u^{2} = 0

Solve

- 8 u + 4 + 3 u^{2} = 0 : u = 2, u = \frac{2}{3}

- 8 u + 4 + 3 u^{2} = 0

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 3, b = - 8, c = 4

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 3 \cdot 4}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 3 \cdot 4}{2 \cdot 3}

(- 8)^{2} - 4 \cdot 3 \cdot 4 = 4

(- 8)^{2} - 4 \cdot 3 \cdot 4

Apply exponent rule:

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

n

is even

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

= 8^{2} - 4 \cdot 3 \cdot 4

Multiply the numbers:

4 \cdot 3 \cdot 4 = 48

= 8^{2} - 48

8^{2} = 64

= 64 - 48

Subtract the numbers:

64 - 48 = 16

= 16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

u_{1, 2} = \frac{- ( - 8 ) \pm 4}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- ( - 8 ) + 4}{2 \cdot 3}, u_{2} = \frac{- ( - 8 ) - 4}{2 \cdot 3}

u = \frac{- ( - 8 ) + 4}{2 \cdot 3} : 2

\frac{- ( - 8 ) + 4}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{8 + 4}{2 \cdot 3}

Add the numbers:

8 + 4 = 12

= \frac{12}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{12}{6}

Divide the numbers:

\frac{12}{6} = 2

= 2

u = \frac{- ( - 8 ) - 4}{2 \cdot 3} : \frac{2}{3}

\frac{- ( - 8 ) - 4}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{8 - 4}{2 \cdot 3}

Subtract the numbers:

8 - 4 = 4

= \frac{4}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{4}{6}

Cancel the common factor:

2

= \frac{2}{3}

The solutions to the quadratic equation are:

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

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

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 8 + \frac{4}{u} + 3 u

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

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

Substitute back

u = sec (x)

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

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

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}{3} :

No Solution

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

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

Popular Examples

3/5 =sin(x)

\frac{3}{5} = sin (x)

(sin^2(x))/(1-cos^2(x))=cot(x)

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

cos^2(x)+cos^3(x)+cos^4(x)+cos^5(x)=0

cos^{2} (x) + cos^{3} (x) + cos^{4} (x) + cos^{5} (x) = 0

sin^2(x)-4sin(x)=-3

sin^{2} (x) - 4 sin (x) = - 3

1+7sinh(x)=4cosh^2(x)

1 + 7 sinh (x) = 4 cosh^{2} (x)

Frequently Asked Questions (FAQ)

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