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

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

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sec(x/2)=sqrt(2)

Solution

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

Solution

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

Degrees

x = 9 0^{\circ} + 72 0^{\circ} n, x = 63 0^{\circ} + 72 0^{\circ} n

Solution steps

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

General solutions for

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

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

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

Solve

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

\frac{x}{2} = \frac{π}{4} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{4} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

2 \cdot \frac{π}{4} + 2 \cdot 2 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn

Solve

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

\frac{x}{2} = \frac{7 π}{4} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{7 π}{4} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

2 \cdot \frac{7 π}{4} + 2 \cdot 2 πn

2 \cdot \frac{7 π}{4} = \frac{7 π}{2}

2 \cdot \frac{7 π}{4}

Multiply fractions:

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

= \frac{7 π 2}{4}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{14 π}{4}

Cancel the common factor:

2

= \frac{7 π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{7 π}{2} + 4 πn

x = \frac{7 π}{2} + 4 πn

x = \frac{7 π}{2} + 4 πn

x = \frac{7 π}{2} + 4 πn

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

Popular Examples

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sin (5 x) = \frac{3}{2}

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sin^{2} (θ) - cos^{2} (θ) = - 1

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solvefor x,sin(x)+cot(x)cos(x)= 1/f solve for

x, sin (x) + cot (x) cos (x) = \frac{1}{f}

1/(sin(x))=2

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

Frequently Asked Questions (FAQ)

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