What is the general solution for 5sec(x)+5tan(x)=5 ?

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

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5sec(x)+5tan(x)=5

Solution

5 sec (x) + 5 tan (x) = 5

Solution

x = 2 πn + 2 π

Degrees

x = 36 0^{\circ} + 36 0^{\circ} n

Solution steps

5 sec (x) + 5 tan (x) = 5

Subtract

5

from both sides

5 sec (x) + 5 tan (x) - 5 = 0

Express with sin, cos

5 \cdot \frac{1}{cos ( x )} + 5 \cdot \frac{sin ( x )}{cos ( x )} - 5 = 0

Simplify

5 \cdot \frac{1}{cos ( x )} + 5 \cdot \frac{sin ( x )}{cos ( x )} - 5 : \frac{5 + 5 sin ( x ) - 5 cos ( x )}{cos ( x )}

5 \cdot \frac{1}{cos ( x )} + 5 \cdot \frac{sin ( x )}{cos ( x )} - 5

5 \cdot \frac{1}{cos ( x )} = \frac{5}{cos ( x )}

5 \cdot \frac{1}{cos ( x )}

Multiply fractions:

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

= \frac{1 \cdot 5}{cos ( x )}

Multiply the numbers:

1 \cdot 5 = 5

= \frac{5}{cos ( x )}

5 \cdot \frac{sin ( x )}{cos ( x )} = \frac{5 sin ( x )}{cos ( x )}

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

Multiply fractions:

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

= \frac{sin ( x ) \cdot 5}{cos ( x )}

= \frac{5}{cos ( x )} + \frac{5 sin ( x )}{cos ( x )} - 5

Combine the fractions

\frac{5}{cos ( x )} + \frac{5 sin ( x )}{cos ( x )} : \frac{5 + 5 sin ( x )}{cos ( x )}

Apply rule

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

= \frac{5 + 5 sin ( x )}{cos ( x )}

= \frac{5 sin ( x ) + 5}{cos ( x )} - 5

Convert element to fraction:

5 = \frac{5 cos ( x )}{cos ( x )}

= \frac{5 + sin ( x ) \cdot 5}{cos ( x )} - \frac{5 cos ( x )}{cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{5 + sin ( x ) \cdot 5 - 5 cos ( x )}{cos ( x )}

\frac{5 + 5 sin ( x ) - 5 cos ( x )}{cos ( x )} = 0

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

5 + 5 sin (x) - 5 cos (x) = 0

Add

5 cos (x)

to both sides

5 + 5 sin (x) = 5 cos (x)

Square both sides

(5 + 5 sin (x))^{2} = (5 cos (x))^{2}

Subtract

(5 cos (x))^{2}

from both sides

(5 + 5 sin (x))^{2} - 25 cos^{2} (x) = 0

Factor

(5 + 5 sin (x))^{2} - 25 cos^{2} (x) : 25 (1 + sin (x) + cos (x)) (1 + sin (x) - cos (x))

(5 + 5 sin (x))^{2} - 25 cos^{2} (x)

Rewrite

(5 + 5 sin (x))^{2} - 25 cos^{2} (x)

(5 + 5 sin (x))^{2} - (5 cos (x))^{2}

(5 + 5 sin (x))^{2} - 25 cos^{2} (x)

Rewrite

25

5^{2}

= (5 + 5 sin (x))^{2} - 5^{2} cos^{2} (x)

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

5^{2} cos^{2} (x) = (5 cos (x))^{2}

= (5 + 5 sin (x))^{2} - (5 cos (x))^{2}

= (5 + 5 sin (x))^{2} - (5 cos (x))^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(5 + 5 sin (x))^{2} - (5 cos (x))^{2} = ((5 + 5 sin (x)) + 5 cos (x)) ((5 + 5 sin (x)) - 5 cos (x))

= ((5 + 5 sin (x)) + 5 cos (x)) ((5 + 5 sin (x)) - 5 cos (x))

Refine

= (5 sin (x) + 5 cos (x) + 5) (5 sin (x) - 5 cos (x) + 5)

Factor

5 + 5 sin (x) + 5 cos (x) : 5 (1 + sin (x) + cos (x))

5 + 5 sin (x) + 5 cos (x)

Factor out common term

5

= 5 (1 + sin (x) + cos (x))

= 5 (sin (x) + cos (x) + 1) (5 sin (x) - 5 cos (x) + 5)

Factor

5 + 5 sin (x) - 5 cos (x) : 5 (1 + sin (x) - cos (x))

5 + 5 sin (x) - 5 cos (x)

Factor out common term

5

= 5 (1 + sin (x) - cos (x))

= 5 (1 + sin (x) + cos (x)) \cdot 5 (1 + sin (x) - cos (x))

Refine

= 25 (1 + sin (x) + cos (x)) (1 + sin (x) - cos (x))

25 (1 + sin (x) + cos (x)) (1 + sin (x) - cos (x)) = 0

Solving each part separately

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

1 + sin (x) + cos (x) = 0 : x = 2 πn + π, x = 2 πn + \frac{3 π}{2}

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

Rewrite using trig identities

1 + sin (x) + cos (x)

sin (x) + cos (x) = 2 sin (x + \frac{π}{4})

sin (x) + cos (x)

Rewrite as

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

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{1}{2}

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (\frac{π}{4}) sin (x) + sin (\frac{π}{4}) cos (x))

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= 2 sin (x + \frac{π}{4})

= 1 + 2 sin (x + \frac{π}{4})

1 + 2 sin (x + \frac{π}{4}) = 0

Move

1

to the right side

1 + 2 sin (x + \frac{π}{4}) = 0

Subtract

1

from both sides

1 + 2 sin (x + \frac{π}{4}) - 1 = 0 - 1

Simplify

2 sin (x + \frac{π}{4}) = - 1

2 sin (x + \frac{π}{4}) = - 1

Divide both sides by

2

2 sin (x + \frac{π}{4}) = - 1

Divide both sides by

2

\frac{2 sin ( x + \frac{π}{4} )}{2} = \frac{- 1}{2}

Simplify

\frac{2 sin ( x + \frac{π}{4} )}{2} = \frac{- 1}{2}

Simplify

\frac{2 sin ( x + \frac{π}{4} )}{2} : sin (x + \frac{π}{4})

\frac{2 sin ( x + \frac{π}{4} )}{2}

Cancel the common factor:

2

= sin (x + \frac{π}{4})

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

sin (x + \frac{π}{4}) = - \frac{2}{2}

sin (x + \frac{π}{4}) = - \frac{2}{2}

sin (x + \frac{π}{4}) = - \frac{2}{2}

General solutions for

sin (x + \frac{π}{4}) = - \frac{2}{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}{4} = \frac{5 π}{4} + 2 πn, x + \frac{π}{4} = \frac{7 π}{4} + 2 πn

x + \frac{π}{4} = \frac{5 π}{4} + 2 πn, x + \frac{π}{4} = \frac{7 π}{4} + 2 πn

Solve

x + \frac{π}{4} = \frac{5 π}{4} + 2 πn : x = 2 πn + π

x + \frac{π}{4} = \frac{5 π}{4} + 2 πn

Move

\frac{π}{4}

to the right side

x + \frac{π}{4} = \frac{5 π}{4} + 2 πn

Subtract

\frac{π}{4}

from both sides

x + \frac{π}{4} - \frac{π}{4} = \frac{5 π}{4} + 2 πn - \frac{π}{4}

Simplify

x + \frac{π}{4} - \frac{π}{4} = \frac{5 π}{4} + 2 πn - \frac{π}{4}

Simplify

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= x

Simplify

\frac{5 π}{4} + 2 πn - \frac{π}{4} : 2 πn + π

\frac{5 π}{4} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{4} + \frac{5 π}{4}

Combine the fractions

- \frac{π}{4} + \frac{5 π}{4} : π

Apply rule

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

= \frac{- π + 5 π}{4}

Add similar elements:

- π + 5 π = 4 π

= \frac{4 π}{4}

Divide the numbers:

\frac{4}{4} = 1

= π

= 2 πn + π

x = 2 πn + π

x = 2 πn + π

x = 2 πn + π

Solve

x + \frac{π}{4} = \frac{7 π}{4} + 2 πn : x = 2 πn + \frac{3 π}{2}

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

Simplify

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= x

Simplify

\frac{7 π}{4} + 2 πn - \frac{π}{4} : 2 πn + \frac{3 π}{2}

\frac{7 π}{4} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{4} + \frac{7 π}{4}

Combine the fractions

- \frac{π}{4} + \frac{7 π}{4} : \frac{3 π}{2}

Apply rule

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

= \frac{- π + 7 π}{4}

Add similar elements:

- π + 7 π = 6 π

= \frac{6 π}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

= 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

x = 2 πn + π, x = 2 πn + \frac{3 π}{2}

1 + sin (x) - cos (x) = 0 : x = 2 πn + \frac{3 π}{2}, x = 2 πn + 2 π

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

Rewrite using trig identities

1 + sin (x) - cos (x)

sin (x) - cos (x) = 2 sin (x - \frac{π}{4})

sin (x) - cos (x)

Rewrite as

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

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{1}{2}

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (\frac{π}{4}) sin (x) - sin (\frac{π}{4}) cos (x))

Use the Angle Sum identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= 2 sin (x - \frac{π}{4})

= 1 + 2 sin (x - \frac{π}{4})

1 + 2 sin (x - \frac{π}{4}) = 0

Move

1

to the right side

1 + 2 sin (x - \frac{π}{4}) = 0

Subtract

1

from both sides

1 + 2 sin (x - \frac{π}{4}) - 1 = 0 - 1

Simplify

2 sin (x - \frac{π}{4}) = - 1

2 sin (x - \frac{π}{4}) = - 1

Divide both sides by

2

2 sin (x - \frac{π}{4}) = - 1

Divide both sides by

2

\frac{2 sin ( x - \frac{π}{4} )}{2} = \frac{- 1}{2}

Simplify

\frac{2 sin ( x - \frac{π}{4} )}{2} = \frac{- 1}{2}

Simplify

\frac{2 sin ( x - \frac{π}{4} )}{2} : sin (x - \frac{π}{4})

\frac{2 sin ( x - \frac{π}{4} )}{2}

Cancel the common factor:

2

= sin (x - \frac{π}{4})

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

sin (x - \frac{π}{4}) = - \frac{2}{2}

sin (x - \frac{π}{4}) = - \frac{2}{2}

sin (x - \frac{π}{4}) = - \frac{2}{2}

General solutions for

sin (x - \frac{π}{4}) = - \frac{2}{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}{4} = \frac{5 π}{4} + 2 πn, x - \frac{π}{4} = \frac{7 π}{4} + 2 πn

x - \frac{π}{4} = \frac{5 π}{4} + 2 πn, x - \frac{π}{4} = \frac{7 π}{4} + 2 πn

Solve

x - \frac{π}{4} = \frac{5 π}{4} + 2 πn : x = 2 πn + \frac{3 π}{2}

x - \frac{π}{4} = \frac{5 π}{4} + 2 πn

Move

\frac{π}{4}

to the right side

x - \frac{π}{4} = \frac{5 π}{4} + 2 πn

Add

\frac{π}{4}

to both sides

x - \frac{π}{4} + \frac{π}{4} = \frac{5 π}{4} + 2 πn + \frac{π}{4}

Simplify

x - \frac{π}{4} + \frac{π}{4} = \frac{5 π}{4} + 2 πn + \frac{π}{4}

Simplify

x - \frac{π}{4} + \frac{π}{4} : x

x - \frac{π}{4} + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} = 0

= x

Simplify

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

\frac{5 π}{4} + 2 πn + \frac{π}{4}

Group like terms

= 2 πn + \frac{π}{4} + \frac{5 π}{4}

Combine the fractions

\frac{π}{4} + \frac{5 π}{4} : \frac{3 π}{2}

Apply rule

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

= \frac{π + 5 π}{4}

Add similar elements:

π + 5 π = 6 π

= \frac{6 π}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

= 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

Solve

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

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

Move

\frac{π}{4}

to the right side

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

Add

\frac{π}{4}

to both sides

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

Simplify

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

Simplify

x - \frac{π}{4} + \frac{π}{4} : x

x - \frac{π}{4} + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} = 0

= x

Simplify

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

\frac{7 π}{4} + 2 πn + \frac{π}{4}

Group like terms

= 2 πn + \frac{π}{4} + \frac{7 π}{4}

Combine the fractions

\frac{π}{4} + \frac{7 π}{4} : 2 π

Apply rule

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

= \frac{π + 7 π}{4}

Add similar elements:

π + 7 π = 8 π

= \frac{8 π}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2 π

= 2 πn + 2 π

x = 2 πn + 2 π

x = 2 πn + 2 π

x = 2 πn + 2 π

x = 2 πn + \frac{3 π}{2}, x = 2 πn + 2 π

Combine all the solutions

x = 2 πn + π, x = 2 πn + \frac{3 π}{2}, x = 2 πn + 2 π

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

5 sec (x) + 5 tan (x) = 5

Remove the ones that don't agree with the equation.

Check the solution

2 πn + π :

False

2 πn + π

Plug in

n = 1

2 π 1 + π

For

5 sec (x) + 5 tan (x) = 5

plug in

x = 2 π 1 + π

5 sec (2 π 1 + π) + 5 tan (2 π 1 + π) = 5

Refine

- 5 = 5

\Rightarrow False

Check the solution

2 πn + \frac{3 π}{2} :

False

2 πn + \frac{3 π}{2}

Plug in

n = 1

2 π 1 + \frac{3 π}{2}

For

5 sec (x) + 5 tan (x) = 5

plug in

x = 2 π 1 + \frac{3 π}{2}

5 sec (2 π 1 + \frac{3 π}{2}) + 5 tan (2 π 1 + \frac{3 π}{2}) = 5

Undefined

\Rightarrow False

Check the solution

2 πn + 2 π :

True

2 πn + 2 π

Plug in

n = 1

2 π 1 + 2 π

For

5 sec (x) + 5 tan (x) = 5

plug in

x = 2 π 1 + 2 π

5 sec (2 π 1 + 2 π) + 5 tan (2 π 1 + 2 π) = 5

Refine

5 = 5

\Rightarrow True

x = 2 πn + 2 π

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Frequently Asked Questions (FAQ)

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