What is the general solution for sqrt(2)sec(4x)-2=0 ?

The general solution for sqrt(2)sec(4x)-2=0 is x= pi/(16)+(pin)/2 ,x=(7pi}{16}+\frac{pin)/2

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sqrt(2)sec(4x)-2=0

Solution

2 sec (4 x) - 2 = 0

Solution

x = \frac{π}{16} + \frac{πn}{2}, x = \frac{7 π}{16} + \frac{πn}{2}

Degrees

x = 11.2 5^{\circ} + 9 0^{\circ} n, x = 78.7 5^{\circ} + 9 0^{\circ} n

Solution steps

2 sec (4 x) - 2 = 0

Move

2

to the right side

2 sec (4 x) - 2 = 0

Add

2

to both sides

2 sec (4 x) - 2 + 2 = 0 + 2

Simplify

2 sec (4 x) = 2

2 sec (4 x) = 2

Divide both sides by

2

2 sec (4 x) = 2

Divide both sides by

2

\frac{2 sec ( 4 x )}{2} = \frac{2}{2}

Simplify

\frac{2 sec ( 4 x )}{2} = \frac{2}{2}

Simplify

\frac{2 sec ( 4 x )}{2} : sec (4 x)

\frac{2 sec ( 4 x )}{2}

Cancel the common factor:

2

= sec (4 x)

Simplify

\frac{2}{2} : 2

\frac{2}{2}

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= 2

sec (4 x) = 2

sec (4 x) = 2

sec (4 x) = 2

General solutions for

sec (4 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}

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

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

Solve

4 x = \frac{π}{4} + 2 πn : x = \frac{π}{16} + \frac{πn}{2}

4 x = \frac{π}{4} + 2 πn

Divide both sides by

4

4 x = \frac{π}{4} + 2 πn

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{\frac{π}{4}}{4} + \frac{2 πn}{4} : \frac{π}{16} + \frac{πn}{2}

\frac{\frac{π}{4}}{4} + \frac{2 πn}{4}

\frac{\frac{π}{4}}{4} = \frac{π}{16}

\frac{\frac{π}{4}}{4}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 4}

Multiply the numbers:

4 \cdot 4 = 16

= \frac{π}{16}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{π}{16} + \frac{πn}{2}

x = \frac{π}{16} + \frac{πn}{2}

x = \frac{π}{16} + \frac{πn}{2}

x = \frac{π}{16} + \frac{πn}{2}

Solve

4 x = \frac{7 π}{4} + 2 πn : x = \frac{7 π}{16} + \frac{πn}{2}

4 x = \frac{7 π}{4} + 2 πn

Divide both sides by

4

4 x = \frac{7 π}{4} + 2 πn

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{\frac{7 π}{4}}{4} + \frac{2 πn}{4} : \frac{7 π}{16} + \frac{πn}{2}

\frac{\frac{7 π}{4}}{4} + \frac{2 πn}{4}

\frac{\frac{7 π}{4}}{4} = \frac{7 π}{16}

\frac{\frac{7 π}{4}}{4}

Apply the fraction rule:

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

= \frac{7 π}{4 \cdot 4}

Multiply the numbers:

4 \cdot 4 = 16

= \frac{7 π}{16}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{7 π}{16} + \frac{πn}{2}

x = \frac{7 π}{16} + \frac{πn}{2}

x = \frac{7 π}{16} + \frac{πn}{2}

x = \frac{7 π}{16} + \frac{πn}{2}

x = \frac{π}{16} + \frac{πn}{2}, x = \frac{7 π}{16} + \frac{πn}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(2)sec(4x)-2=0 ?
The general solution for sqrt(2)sec(4x)-2=0 is x= pi/(16)+(pin)/2 ,x=(7pi}{16}+\frac{pin)/2