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

The general solution for sec(x)=2tan(x) is x= pi/6+2pin,x=(5pi)/6+2pin

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sec(x)=2tan(x)

sec (x) = 2 tan (x)

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

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

sec (x) = 2 tan (x)

Subtract

2 tan (x)

from both sides

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

Express with sin, cos

sec (x) - 2 tan (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply

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

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

Multiply fractions:

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

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

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

Apply rule

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

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

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

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

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

1 - 2 sin (x) = 0

Move

1

to the right side

1 - 2 sin (x) = 0

Subtract

1

from both sides

1 - 2 sin (x) - 1 = 0 - 1

Simplify

- 2 sin (x) = - 1

- 2 sin (x) = - 1

Divide both sides by

- 2

- 2 sin (x) = - 1

Divide both sides by

- 2

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

Simplify

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

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

General solutions for

sin (x) = \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}

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

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

cos^{2} (a) = 2

arctan (2 x) = - \frac{16}{10}

3 tan (x) + 4 = 0

x, sin (x) = - \frac{3}{2}

\frac{7 π}{2} = 7 - 3 sin (x)

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