What is the general solution for 2tan((pix)/2)=sec((pix)/2) ?

The general solution for 2tan((pix)/2)=sec((pix)/2) is x=(1+12n)/3 ,x=(5+12n)/3

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2tan((pix)/2)=sec((pix)/2)

Solution

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

Solution

x = \frac{1 + 12 n}{3}, x = \frac{5 + 12 n}{3}

Degrees

x = 19.09859 \dots^{\circ} + 229.18311 \dots^{\circ} n, x = 95.49296 \dots^{\circ} + 229.18311 \dots^{\circ} n

Solution steps

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

Subtract

sec (\frac{π x}{2})

from both sides

2 tan (\frac{π x}{2}) - sec (\frac{π x}{2}) = 0

Let:

u = \frac{π x}{2}

2 tan (u) - sec (u) = 0

Express with sin, cos

- sec (u) + 2 tan (u)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= - \frac{1}{cos ( u )} + 2 tan (u)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= - \frac{1}{cos ( u )} + 2 \cdot \frac{sin ( u )}{cos ( u )}

Simplify

- \frac{1}{cos ( u )} + 2 \cdot \frac{sin ( u )}{cos ( u )} : \frac{- 1 + 2 sin ( u )}{cos ( u )}

- \frac{1}{cos ( u )} + 2 \cdot \frac{sin ( u )}{cos ( u )}

Multiply

2 \cdot \frac{sin ( u )}{cos ( u )} : \frac{2 sin ( u )}{cos ( u )}

2 \cdot \frac{sin ( u )}{cos ( u )}

Multiply fractions:

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

= \frac{sin ( u ) \cdot 2}{cos ( u )}

= - \frac{1}{cos ( u )} + \frac{2 sin ( u )}{cos ( u )}

Apply rule

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

= \frac{- 1 + 2 sin ( u )}{cos ( u )}

= \frac{- 1 + 2 sin ( u )}{cos ( u )}

\frac{- 1 + 2 sin ( u )}{cos ( u )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- 1 + 2 sin (u) = 0

Move

1

to the right side

- 1 + 2 sin (u) = 0

Add

1

to both sides

- 1 + 2 sin (u) + 1 = 0 + 1

Simplify

2 sin (u) = 1

2 sin (u) = 1

Divide both sides by

2

2 sin (u) = 1

Divide both sides by

2

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

Simplify

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

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

General solutions for

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

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

u = \frac{π}{6} + 2 πn, u = \frac{5 π}{6} + 2 πn

u = \frac{π}{6} + 2 πn, u = \frac{5 π}{6} + 2 πn

Substitute back

u = \frac{π x}{2}

\frac{π x}{2} = \frac{π}{6} + 2 πn : x = \frac{1 + 12 n}{3}

\frac{π x}{2} = \frac{π}{6} + 2 πn

Multiply both sides by

2

\frac{π x}{2} = \frac{π}{6} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 π x}{2} : π x

\frac{2 π x}{2}

Divide the numbers:

\frac{2}{2} = 1

= π x

Simplify

2 \cdot \frac{π}{6} + 2 \cdot 2 πn : \frac{π}{3} + 4 πn

2 \cdot \frac{π}{6} + 2 \cdot 2 πn

2 \cdot \frac{π}{6} = \frac{π}{3}

2 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 2}{6}

Cancel the common factor:

2

= \frac{π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{π}{3} + 4 πn

π x = \frac{π}{3} + 4 πn

π x = \frac{π}{3} + 4 πn

π x = \frac{π}{3} + 4 πn

Divide both sides by

π

π x = \frac{π}{3} + 4 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{\frac{π}{3}}{π} + \frac{4 πn}{π} : \frac{1 + 12 n}{3}

\frac{\frac{π}{3}}{π} + \frac{4 πn}{π}

Apply rule

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

= \frac{\frac{π}{3} + 4 πn}{π}

Join

\frac{π}{3} + 4 πn : \frac{π + 12 πn}{3}

\frac{π}{3} + 4 πn

Convert element to fraction:

4 πn = \frac{4 πn 3}{3}

= \frac{π}{3} + \frac{4 πn \cdot 3}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{π + 4 πn \cdot 3}{3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{π + 12 πn}{3}

= \frac{\frac{π + 12 πn}{3}}{π}

Apply the fraction rule:

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

= \frac{π + 12 πn}{3 π}

Factor out common term

π

= \frac{π ( 1 + 12 n )}{3 π}

Cancel the common factor:

π

= \frac{1 + 12 n}{3}

x = \frac{1 + 12 n}{3}

x = \frac{1 + 12 n}{3}

x = \frac{1 + 12 n}{3}

\frac{π x}{2} = \frac{5 π}{6} + 2 πn : x = \frac{5 + 12 n}{3}

\frac{π x}{2} = \frac{5 π}{6} + 2 πn

Multiply both sides by

2

\frac{π x}{2} = \frac{5 π}{6} + 2 πn

Multiply both sides by

2

\frac{2 π x}{2} = 2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 π x}{2} = 2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 π x}{2} : π x

\frac{2 π x}{2}

Divide the numbers:

\frac{2}{2} = 1

= π x

Simplify

2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn : \frac{5 π}{3} + 4 πn

2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{5 π}{6} = \frac{5 π}{3}

2 \cdot \frac{5 π}{6}

Multiply fractions:

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

= \frac{5 π 2}{6}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{6}

Cancel the common factor:

2

= \frac{5 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{5 π}{3} + 4 πn

π x = \frac{5 π}{3} + 4 πn

π x = \frac{5 π}{3} + 4 πn

π x = \frac{5 π}{3} + 4 πn

Divide both sides by

π

π x = \frac{5 π}{3} + 4 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{\frac{5 π}{3}}{π} + \frac{4 πn}{π} : \frac{5 + 12 n}{3}

\frac{\frac{5 π}{3}}{π} + \frac{4 πn}{π}

Apply rule

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

= \frac{\frac{5 π}{3} + 4 πn}{π}

Join

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

\frac{5 π}{3} + 4 πn

Convert element to fraction:

4 πn = \frac{4 πn 3}{3}

= \frac{5 π}{3} + \frac{4 πn \cdot 3}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{5 π + 4 πn \cdot 3}{3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{5 π + 12 πn}{3}

= \frac{\frac{5 π + 12 πn}{3}}{π}

Apply the fraction rule:

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

= \frac{5 π + 12 πn}{3 π}

Factor out common term

π

= \frac{π ( 5 + 12 n )}{3 π}

Cancel the common factor:

π

= \frac{5 + 12 n}{3}

x = \frac{5 + 12 n}{3}

x = \frac{5 + 12 n}{3}

x = \frac{5 + 12 n}{3}

x = \frac{1 + 12 n}{3}, x = \frac{5 + 12 n}{3}

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Frequently Asked Questions (FAQ)

What is the general solution for 2tan((pix)/2)=sec((pix)/2) ?
The general solution for 2tan((pix)/2)=sec((pix)/2) is x=(1+12n)/3 ,x=(5+12n)/3