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

The general solution for 2sec(x)=sqrt(2)tan(x)+3sec(x) is x=(5pi)/4+2pin,x=(7pi)/4+2pin

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Popular >

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

Solution

2 sec (x) = 2 tan (x) + 3 sec (x)

Solution

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Degrees

x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

2 sec (x) = 2 tan (x) + 3 sec (x)

Subtract

2 tan (x) + 3 sec (x)

from both sides

- sec (x) - 2 tan (x) = 0

Express with sin, cos

- sec (x) - 2 tan (x)

Use the basic trigonometric identity:

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

= - \frac{1}{cos ( x )} - 2 tan (x)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

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

Simplify

- \frac{1}{cos ( x )} - 2 \frac{sin ( x )}{cos ( x )} : \frac{- 1 - 2 sin ( x )}{cos ( x )}

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

Multiply

2 \frac{sin ( x )}{cos ( x )} : \frac{2 sin ( x )}{cos ( x )}

2 \frac{sin ( x )}{cos ( x )}

Multiply fractions:

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

= \frac{sin ( x ) 2}{cos ( x )}

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

Apply rule

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

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

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

\frac{- 1 - sin ( x ) 2}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- 1 - sin (x) 2 = 0

Move

1

to the right side

- 1 - sin (x) 2 = 0

Add

1

to both sides

- 1 - sin (x) 2 + 1 = 0 + 1

Simplify

- sin (x) 2 = 1

- sin (x) 2 = 1

Divide both sides by

- 2

- sin (x) 2 = 1

Divide both sides by

- 2

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

Simplify

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

Simplify

\frac{- sin ( x ) 2}{- 2} : sin (x)

\frac{- sin ( x ) 2}{- 2}

Apply the fraction rule:

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

= \frac{sin ( x ) 2}{2}

Cancel the common factor:

2

= sin (x)

Simplify

\frac{1}{- 2} : - \frac{2}{2}

\frac{1}{- 2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

sin (x) = - \frac{2}{2}

sin (x) = - \frac{2}{2}

sin (x) = - \frac{2}{2}

General solutions for

sin (x) = - \frac{2}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Popular Examples

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4 cos (x) + 3 sin (x) = 5

sin(x)-csc(x)+1=0

sin (x) - csc (x) + 1 = 0

tan^2(θ)-sec^2(θ)=cos(-θ)

tan^{2} (θ) - sec^{2} (θ) = cos (- θ)

-2sin(t)-4cos(2t)=0

- 2 sin (t) - 4 cos (2 t) = 0

8sin(2x)-4sqrt(3)=0

8 sin (2 x) - 43 = 0

Frequently Asked Questions (FAQ)

What is the general solution for 2sec(x)=sqrt(2)tan(x)+3sec(x) ?
The general solution for 2sec(x)=sqrt(2)tan(x)+3sec(x) is x=(5pi)/4+2pin,x=(7pi)/4+2pin