What is the general solution for sec(x+pi/4)=-sqrt(2),-pi<= x<= pi ?

Q: What is the general solution for sec(x+pi/4)=-sqrt(2),-pi<= x<= pi ?

The general solution for sec(x+pi/4)=-sqrt(2),-pi<= x<= pi is x= pi/2 ,x=pi,x=-pi

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sec(x+pi/4)=-sqrt(2),-pi<= x<= pi

Solution

sec (x + \frac{π}{4}) = - 2, - π \leq x \leq π

Solution

x = \frac{π}{2}, x = π, x = - π

Degrees

x = 9 0^{\circ}, x = 18 0^{\circ}, x = - 18 0^{\circ}

Solution steps

sec (x + \frac{π}{4}) = - 2, - π \leq x \leq π

General solutions for

sec (x + \frac{π}{4}) = - 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} = \frac{3 π}{4} + 2 πn, x + \frac{π}{4} = \frac{5 π}{4} + 2 πn

x + \frac{π}{4} = \frac{3 π}{4} + 2 πn, x + \frac{π}{4} = \frac{5 π}{4} + 2 πn

Solve

x + \frac{π}{4} = \frac{3 π}{4} + 2 πn : x = 2 πn + \frac{π}{2}

x + \frac{π}{4} = \frac{3 π}{4} + 2 πn

Move

\frac{π}{4}

to the right side

x + \frac{π}{4} = \frac{3 π}{4} + 2 πn

Subtract

\frac{π}{4}

from both sides

x + \frac{π}{4} - \frac{π}{4} = \frac{3 π}{4} + 2 πn - \frac{π}{4}

Simplify

x + \frac{π}{4} - \frac{π}{4} = \frac{3 π}{4} + 2 πn - \frac{π}{4}

Simplify

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= x

Simplify

\frac{3 π}{4} + 2 πn - \frac{π}{4} : 2 πn + \frac{π}{2}

\frac{3 π}{4} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{4} + \frac{3 π}{4}

Combine the fractions

- \frac{π}{4} + \frac{3 π}{4} : \frac{π}{2}

Apply rule

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

= \frac{- π + 3 π}{4}

Add similar elements:

- π + 3 π = 2 π

= \frac{2 π}{4}

Cancel the common factor:

2

= \frac{π}{2}

= 2 πn + \frac{π}{2}

x = 2 πn + \frac{π}{2}

x = 2 πn + \frac{π}{2}

x = 2 πn + \frac{π}{2}

Solve

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

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

Simplify

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= x

Simplify

\frac{5 π}{4} + 2 πn - \frac{π}{4} : 2 πn + π

\frac{5 π}{4} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{4} + \frac{5 π}{4}

Combine the fractions

- \frac{π}{4} + \frac{5 π}{4} : π

Apply rule

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

= \frac{- π + 5 π}{4}

Add similar elements:

- π + 5 π = 4 π

= \frac{4 π}{4}

Divide the numbers:

\frac{4}{4} = 1

= π

= 2 πn + π

x = 2 πn + π

x = 2 πn + π

x = 2 πn + π

x = 2 πn + \frac{π}{2}, x = 2 πn + π

Solutions for the range

- π \leq x \leq π

x = \frac{π}{2}, x = π, x = - π

Popular Examples

(56)/((7.071*7.681))=cos(θ)

\frac{56}{( 7.071 \cdot 7.681 )} = cos (θ)

tan(2x+10)=cot(x-40)

tan (2 x + 1 0^{\circ}) = cot (x - 4 0^{\circ})

6cos(2t)=0

6 cos (2 t) = 0

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2 cos^{2} (t) - 3 cos (t) = - 1, 0 \leq t \leq 2 π

4cos(4x)=3

4 cos (4 x) = 3

Frequently Asked Questions (FAQ)

What is the general solution for sec(x+pi/4)=-sqrt(2),-pi<= x<= pi ?
The general solution for sec(x+pi/4)=-sqrt(2),-pi<= x<= pi is x= pi/2 ,x=pi,x=-pi