What is the general solution for sqrt(3)sec(4x)=2 ?

The general solution for sqrt(3)sec(4x)=2 is x= pi/(24)+(pin)/2 ,x=(11pi}{24}+\frac{pin)/2

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sqrt(3)sec(4x)=2

Solution

3 sec (4 x) = 2

Solution

x = \frac{π}{24} + \frac{πn}{2}, x = \frac{11 π}{24} + \frac{πn}{2}

Degrees

x = 7. 5^{\circ} + 9 0^{\circ} n, x = 82. 5^{\circ} + 9 0^{\circ} n

Solution steps

3 sec (4 x) = 2

Divide both sides by

3

3 sec (4 x) = 2

Divide both sides by

3

\frac{3 sec ( 4 x )}{3} = \frac{2}{3}

Simplify

\frac{3 sec ( 4 x )}{3} = \frac{2}{3}

Simplify

\frac{3 sec ( 4 x )}{3} : sec (4 x)

\frac{3 sec ( 4 x )}{3}

Cancel the common factor:

3

= sec (4 x)

Simplify

\frac{2}{3} : \frac{2 3}{3}

\frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{2 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{2 3}{3}

sec (4 x) = \frac{2 3}{3}

sec (4 x) = \frac{2 3}{3}

sec (4 x) = \frac{2 3}{3}

General solutions for

sec (4 x) = \frac{2 3}{3}

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}{6} + 2 πn, 4 x = \frac{11 π}{6} + 2 πn

4 x = \frac{π}{6} + 2 πn, 4 x = \frac{11 π}{6} + 2 πn

Solve

4 x = \frac{π}{6} + 2 πn : x = \frac{π}{24} + \frac{πn}{2}

4 x = \frac{π}{6} + 2 πn

Divide both sides by

4

4 x = \frac{π}{6} + 2 πn

Divide both sides by

4

\frac{4 x}{4} = \frac{\frac{π}{6}}{4} + \frac{2 πn}{4}

Simplify

\frac{4 x}{4} = \frac{\frac{π}{6}}{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}{6}}{4} + \frac{2 πn}{4} : \frac{π}{24} + \frac{πn}{2}

\frac{\frac{π}{6}}{4} + \frac{2 πn}{4}

\frac{\frac{π}{6}}{4} = \frac{π}{24}

\frac{\frac{π}{6}}{4}

Apply the fraction rule:

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

= \frac{π}{6 \cdot 4}

Multiply the numbers:

6 \cdot 4 = 24

= \frac{π}{24}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{π}{24} + \frac{πn}{2}

x = \frac{π}{24} + \frac{πn}{2}

x = \frac{π}{24} + \frac{πn}{2}

x = \frac{π}{24} + \frac{πn}{2}

Solve

4 x = \frac{11 π}{6} + 2 πn : x = \frac{11 π}{24} + \frac{πn}{2}

4 x = \frac{11 π}{6} + 2 πn

Divide both sides by

4

4 x = \frac{11 π}{6} + 2 πn

Divide both sides by

4

\frac{4 x}{4} = \frac{\frac{11 π}{6}}{4} + \frac{2 πn}{4}

Simplify

\frac{4 x}{4} = \frac{\frac{11 π}{6}}{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{11 π}{6}}{4} + \frac{2 πn}{4} : \frac{11 π}{24} + \frac{πn}{2}

\frac{\frac{11 π}{6}}{4} + \frac{2 πn}{4}

\frac{\frac{11 π}{6}}{4} = \frac{11 π}{24}

\frac{\frac{11 π}{6}}{4}

Apply the fraction rule:

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

= \frac{11 π}{6 \cdot 4}

Multiply the numbers:

6 \cdot 4 = 24

= \frac{11 π}{24}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{11 π}{24} + \frac{πn}{2}

x = \frac{11 π}{24} + \frac{πn}{2}

x = \frac{11 π}{24} + \frac{πn}{2}

x = \frac{11 π}{24} + \frac{πn}{2}

x = \frac{π}{24} + \frac{πn}{2}, x = \frac{11 π}{24} + \frac{πn}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(3)sec(4x)=2 ?
The general solution for sqrt(3)sec(4x)=2 is x= pi/(24)+(pin)/2 ,x=(11pi}{24}+\frac{pin)/2