What is the general solution for 2cos(x)+2sqrt(2)=3sec(x) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

2cos(x)+2sqrt(2)=3sec(x)

Solution

2 cos (x) + 22 = 3 sec (x)

Solution

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

2 cos (x) + 22 = 3 sec (x)

Subtract

3 sec (x)

from both sides

2 cos (x) + 22 - 3 sec (x) = 0

Rewrite using trig identities

2 cos (x) + 22 - 3 sec (x)

Use the basic trigonometric identity:

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

= 2 \cdot \frac{1}{sec ( x )} + 22 - 3 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 )}

= \frac{2}{sec ( x )} + 22 - 3 sec (x)

\frac{2}{sec ( x )} + 22 - 3 sec (x) = 0

Solve by substitution

\frac{2}{sec ( x )} + 22 - 3 sec (x) = 0

Let:

sec (x) = u

\frac{2}{u} + 22 - 3 u = 0

\frac{2}{u} + 22 - 3 u = 0 : u = - \frac{2}{3}, u = 2

\frac{2}{u} + 22 - 3 u = 0

Multiply both sides by

u

\frac{2}{u} + 22 - 3 u = 0

Multiply both sides by

u

\frac{2}{u} u + 22 u - 3 uu = 0 \cdot u

Simplify

\frac{2}{u} u + 22 u - 3 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

- 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

2 + 22 u - 3 u^{2} = 0

2 + 22 u - 3 u^{2} = 0

2 + 22 u - 3 u^{2} = 0

Solve

2 + 22 u - 3 u^{2} = 0 : u = - \frac{2}{3}, u = 2

2 + 22 u - 3 u^{2} = 0

Write in the standard form

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

- 3 u^{2} + 22 u + 2 = 0

Solve with the quadratic formula

- 3 u^{2} + 22 u + 2 = 0

Quadratic Equation Formula:

For

a = - 3, b = 22, c = 2

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

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

(22)^{2} - 4 (- 3) \cdot 2 = 42

(22)^{2} - 4 (- 3) \cdot 2

Apply rule

- (- a) = a

= (22)^{2} + 4 \cdot 3 \cdot 2

(22)^{2} = 2^{3}

(22)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} (2)^{2}

(2)^{2} : 2

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2

= 2^{2} \cdot 2

Apply exponent rule:

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

2^{2} \cdot 2 = 2^{2 + 1}

= 2^{2 + 1}

Add the numbers:

2 + 1 = 3

= 2^{3}

4 \cdot 3 \cdot 2 = 24

4 \cdot 3 \cdot 2

Multiply the numbers:

4 \cdot 3 \cdot 2 = 24

= 24

= 2^{3} + 24

2^{3} = 8

= 8 + 24

Add the numbers:

8 + 24 = 32

= 32

Prime factorization of

32 : 2^{5}

32

32

divides by

232 = 16 \cdot 2

= 2 \cdot 16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

= 2^{5}

= 2^{5}

Apply exponent rule:

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

= 2^{4} \cdot 2

Apply radical rule:

= 2 2^{4}

Apply radical rule:

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 2

Refine

= 42

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

Separate the solutions

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

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

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

Remove parentheses:

(- a) = - a

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

Add similar elements:

- 22 + 42 = 22

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{2 2}{- 6}

Apply the fraction rule:

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

= - \frac{2 2}{6}

Cancel the common factor:

2

= - \frac{2}{3}

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

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

Remove parentheses:

(- a) = - a

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

Add similar elements:

- 22 - 42 = - 62

= \frac{- 6 2}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 6 2}{- 6}

Apply the fraction rule:

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

= \frac{6 2}{6}

Divide the numbers:

\frac{6}{6} = 1

= 2

The solutions to the quadratic equation are:

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

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

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{2}{u} + 22 - 3 u

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

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

Substitute back

u = sec (x)

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

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

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

No Solution

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

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

No Solution

sec (x) = 2 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 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}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Combine all the solutions

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Popular Examples

cot^2(x)-3cot(x)-2=0

cot^{2} (x) - 3 cot (x) - 2 = 0

cos(x-75)= 1/2

cos (x - 7 5^{\circ}) = \frac{1}{2}

sin(2x-23)=-(sqrt(2))/2

sin (2 x - 2 3^{\circ}) = - \frac{2}{2}

tan^2(x)-sec(x)=-1

tan^{2} (x) - sec (x) = - 1

sin^2(x)=((10m-5))/2

sin^{2} (x) = \frac{( 10 m - 5 )}{2}

Frequently Asked Questions (FAQ)

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