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

The general solution for sec(2x)+tan(2x)=2 is x=0.32175…+pin

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Popular Trigonometry >

sec(2x)+tan(2x)=2

Solution

sec (2 x) + tan (2 x) = 2

Solution

x = 0.32175 \dots + πn

Degrees

x = 18.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

sec (2 x) + tan (2 x) = 2

Subtract

2

from both sides

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

Express with sin, cos

- 2 + sec (2 x) + tan (2 x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Combine the fractions

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

Apply rule

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

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

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

Rewrite using trig identities

1 + sin (2 x) - 2 cos (2 x)

Use the Double Angle identity:

cos (2 x) = cos^{2} (x) - sin^{2} (x)

= 1 + sin (2 x) - 2 (cos^{2} (x) - sin^{2} (x))

1 + sin (2 x) = (sin (x) + cos (x))^{2}

1 + sin (2 x)

Use the Pythagorean identity:

1 = cos^{2} (x) + sin^{2} (x)

= (cos^{2} (x) + sin^{2} (x)) + sin (2 x)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= cos^{2} (x) + sin^{2} (x) + 2 sin (x) cos (x)

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x) = (sin (x) + cos (x))^{2}

= (sin (x) + cos (x))^{2}

= (sin (x) + cos (x))^{2} - 2 (cos^{2} (x) - sin^{2} (x))

Simplify

(sin (x) + cos (x))^{2} - 2 (cos^{2} (x) - sin^{2} (x)) : 3 sin^{2} (x) + 2 sin (x) cos (x) - cos^{2} (x)

(sin (x) + cos (x))^{2} - 2 (cos^{2} (x) - sin^{2} (x))

(sin (x) + cos (x))^{2} : sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x)

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = sin (x), b = cos (x)

= sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x)

= sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x) - 2 (cos^{2} (x) - sin^{2} (x))

Expand

- 2 (cos^{2} (x) - sin^{2} (x)) : - 2 cos^{2} (x) + 2 sin^{2} (x)

- 2 (cos^{2} (x) - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 2, b = cos^{2} (x), c = sin^{2} (x)

= - 2 cos^{2} (x) - (- 2) sin^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 2 cos^{2} (x) + 2 sin^{2} (x)

= sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x) - 2 cos^{2} (x) + 2 sin^{2} (x)

Simplify

sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x) - 2 cos^{2} (x) + 2 sin^{2} (x) : 3 sin^{2} (x) + 2 sin (x) cos (x) - cos^{2} (x)

sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x) - 2 cos^{2} (x) + 2 sin^{2} (x)

Add similar elements:

cos^{2} (x) - 2 cos^{2} (x) = - cos^{2} (x)

= sin^{2} (x) + 2 sin (x) cos (x) - cos^{2} (x) + 2 sin^{2} (x)

Add similar elements:

sin^{2} (x) + 2 sin^{2} (x) = 3 sin^{2} (x)

= 3 sin^{2} (x) + 2 sin (x) cos (x) - cos^{2} (x)

= 3 sin^{2} (x) + 2 sin (x) cos (x) - cos^{2} (x)

= 3 sin^{2} (x) + 2 sin (x) cos (x) - cos^{2} (x)

- cos^{2} (x) + 3 sin^{2} (x) + 2 cos (x) sin (x) = 0

Factor

- cos^{2} (x) + 3 sin^{2} (x) + 2 cos (x) sin (x) : (3 sin (x) - cos (x)) (sin (x) + cos (x))

- cos^{2} (x) + 3 sin^{2} (x) + 2 cos (x) sin (x)

Break the expression into groups

3 sin^{2} (x) + 2 sin (x) cos (x) - cos^{2} (x)

Definition

Factors of

3 : 1, 3

3

Divisors (Factors)

Find the Prime factors of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Add 1

1

The factors of

3

1, 3

Negative factors of

3 : - 1, - 3

Multiply the factors by

- 1

to get the negative factors

- 1, - 3

For every two factors such that

u * v = - 3,

check if

u + v = 2

Check

u = 1, v = - 3 : u * v = - 3, u + v = - 2 \Rightarrow

FalseCheck

u = 3, v = - 1 : u * v = - 3, u + v = 2 \Rightarrow

True

u = 3, v = - 1

Group into

(a x^{2} + ux y) + (vx y + c y^{2})

(3 sin^{2} (x) - sin (x) cos (x)) + (3 sin (x) cos (x) - cos^{2} (x))

= (3 sin^{2} (x) - sin (x) cos (x)) + (3 sin (x) cos (x) - cos^{2} (x))

Factor out

sin (x)

from

3 sin^{2} (x) - sin (x) cos (x) : sin (x) (3 sin (x) - cos (x))

3 sin^{2} (x) - sin (x) cos (x)

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

sin^{2} (x) = sin (x) sin (x)

= 3 sin (x) sin (x) - sin (x) cos (x)

Factor out common term

sin (x)

= sin (x) (3 sin (x) - cos (x))

Factor out

cos (x)

from

3 sin (x) cos (x) - cos^{2} (x) : cos (x) (3 sin (x) - cos (x))

3 sin (x) cos (x) - cos^{2} (x)

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

cos^{2} (x) = cos (x) cos (x)

= 3 sin (x) cos (x) - cos (x) cos (x)

Factor out common term

cos (x)

= cos (x) (3 sin (x) - cos (x))

= sin (x) (3 sin (x) - cos (x)) + cos (x) (3 sin (x) - cos (x))

Factor out common term

3 sin (x) - cos (x)

= (3 sin (x) - cos (x)) (sin (x) + cos (x))

(3 sin (x) - cos (x)) (sin (x) + cos (x)) = 0

Solving each part separately

3 sin (x) - cos (x) = 0 or sin (x) + cos (x) = 0

3 sin (x) - cos (x) = 0 : x = arctan (\frac{1}{3}) + πn

3 sin (x) - cos (x) = 0

Rewrite using trig identities

3 sin (x) - cos (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{3 sin ( x ) - cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

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

Use the basic trigonometric identity:

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

3 tan (x) - 1 = 0

3 tan (x) - 1 = 0

Move

1

to the right side

3 tan (x) - 1 = 0

Add

1

to both sides

3 tan (x) - 1 + 1 = 0 + 1

Simplify

3 tan (x) = 1

3 tan (x) = 1

Divide both sides by

3

3 tan (x) = 1

Divide both sides by

3

\frac{3 tan ( x )}{3} = \frac{1}{3}

Simplify

tan (x) = \frac{1}{3}

tan (x) = \frac{1}{3}

Apply trig inverse properties

tan (x) = \frac{1}{3}

General solutions for

tan (x) = \frac{1}{3}

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{1}{3}) + πn

x = arctan (\frac{1}{3}) + πn

sin (x) + cos (x) = 0 : x = \frac{3 π}{4} + πn

sin (x) + cos (x) = 0

Rewrite using trig identities

sin (x) + cos (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{sin ( x ) + cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

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

Use the basic trigonometric identity:

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

tan (x) + 1 = 0

tan (x) + 1 = 0

Move

1

to the right side

tan (x) + 1 = 0

Subtract

1

from both sides

tan (x) + 1 - 1 = 0 - 1

Simplify

tan (x) = - 1

tan (x) = - 1

General solutions for

tan (x) = - 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

Combine all the solutions

x = arctan (\frac{1}{3}) + πn, x = \frac{3 π}{4} + πn

Since the equation is undefined for:

\frac{3 π}{4} + πn

x = arctan (\frac{1}{3}) + πn

Show solutions in decimal form

x = 0.32175 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sec(2x)+tan(2x)=2 ?
The general solution for sec(2x)+tan(2x)=2 is x=0.32175…+pin