What is the general solution for -6=-4+sec(x+pi/4) ?

The general solution for -6=-4+sec(x+pi/4) is x=2pin+(5pi)/(12),x=2pin+(13pi)/(12)

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-6=-4+sec(x+pi/4)

Solution

- 6 = - 4 + sec (x + \frac{π}{4})

Solution

x = 2 πn + \frac{5 π}{12}, x = 2 πn + \frac{13 π}{12}

Degrees

x = 7 5^{\circ} + 36 0^{\circ} n, x = 19 5^{\circ} + 36 0^{\circ} n

Solution steps

- 6 = - 4 + sec (x + \frac{π}{4})

Switch sides

- 4 + sec (x + \frac{π}{4}) = - 6

Move

4

to the right side

- 4 + sec (x + \frac{π}{4}) = - 6

Add

4

to both sides

- 4 + sec (x + \frac{π}{4}) + 4 = - 6 + 4

Simplify

sec (x + \frac{π}{4}) = - 2

sec (x + \frac{π}{4}) = - 2

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

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

Solve

x + \frac{π}{4} = \frac{2 π}{3} + 2 πn : x = 2 πn + \frac{5 π}{12}

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

x + \frac{π}{4} - \frac{π}{4} = \frac{2 π}{3} + 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{2 π}{3} + 2 πn - \frac{π}{4} : 2 πn + \frac{5 π}{12}

\frac{2 π}{3} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{4} + \frac{2 π}{3}

Least Common Multiplier of

4, 3 : 12

4, 3

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

4

3

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

For

\frac{2 π}{3} :

multiply the denominator and numerator by

4

\frac{2 π}{3} = \frac{2 π 4}{3 \cdot 4} = \frac{8 π}{12}

= - \frac{π 3}{12} + \frac{8 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 3 + 8 π}{12}

Add similar elements:

- 3 π + 8 π = 5 π

= 2 πn + \frac{5 π}{12}

x = 2 πn + \frac{5 π}{12}

x = 2 πn + \frac{5 π}{12}

x = 2 πn + \frac{5 π}{12}

Solve

x + \frac{π}{4} = \frac{4 π}{3} + 2 πn : x = 2 πn + \frac{13 π}{12}

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

x + \frac{π}{4} - \frac{π}{4} = \frac{4 π}{3} + 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{4 π}{3} + 2 πn - \frac{π}{4} : 2 πn + \frac{13 π}{12}

\frac{4 π}{3} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{4} + \frac{4 π}{3}

Least Common Multiplier of

4, 3 : 12

4, 3

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

4

3

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

For

\frac{4 π}{3} :

multiply the denominator and numerator by

4

\frac{4 π}{3} = \frac{4 π 4}{3 \cdot 4} = \frac{16 π}{12}

= - \frac{π 3}{12} + \frac{16 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 3 + 16 π}{12}

Add similar elements:

- 3 π + 16 π = 13 π

= 2 πn + \frac{13 π}{12}

x = 2 πn + \frac{13 π}{12}

x = 2 πn + \frac{13 π}{12}

x = 2 πn + \frac{13 π}{12}

x = 2 πn + \frac{5 π}{12}, x = 2 πn + \frac{13 π}{12}

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Frequently Asked Questions (FAQ)

What is the general solution for -6=-4+sec(x+pi/4) ?
The general solution for -6=-4+sec(x+pi/4) is x=2pin+(5pi)/(12),x=2pin+(13pi)/(12)