What is the general solution for (cos(θ))/(1+sin(θ))=tan(pi/4-θ) ?

The general solution for (cos(θ))/(1+sin(θ))=tan(pi/4-θ) is θ=2pin+2pi

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(cos(θ))/(1+sin(θ))=tan(pi/4-θ)

Solution

\frac{cos ( θ )}{1 + sin ( θ )} = tan (\frac{π}{4} - θ)

Solution

θ = 2 πn + 2 π

Degrees

θ = 36 0^{\circ} + 36 0^{\circ} n

Solution steps

\frac{cos ( θ )}{1 + sin ( θ )} = tan (\frac{π}{4} - θ)

Rewrite using trig identities

\frac{cos ( θ )}{1 + sin ( θ )} = tan (\frac{π}{4} - θ)

Rewrite using trig identities

tan (\frac{π}{4} - θ)

Use the basic trigonometric identity:

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

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

Use the Angle Difference identity:

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

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

Use the Angle Difference identity:

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

= \frac{sin ( \frac{π}{4} ) cos ( θ ) - cos ( \frac{π}{4} ) sin ( θ )}{cos ( \frac{π}{4} ) cos ( θ ) + sin ( \frac{π}{4} ) sin ( θ )}

Simplify

\frac{sin ( \frac{π}{4} ) cos ( θ ) - cos ( \frac{π}{4} ) sin ( θ )}{cos ( \frac{π}{4} ) cos ( θ ) + sin ( \frac{π}{4} ) sin ( θ )} : \frac{cos ( θ ) - sin ( θ )}{cos ( θ ) + sin ( θ )}

\frac{sin ( \frac{π}{4} ) cos ( θ ) - cos ( \frac{π}{4} ) sin ( θ )}{cos ( \frac{π}{4} ) cos ( θ ) + sin ( \frac{π}{4} ) sin ( θ )}

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

sin (\frac{π}{4}) cos (θ) - cos (\frac{π}{4}) sin (θ)

Simplify

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Use the following trivial identity:

sin (\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}

= \frac{2}{2}

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

Simplify

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

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

= \frac{2}{2}

= \frac{2}{2} cos (θ) - \frac{2}{2} sin (θ)

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

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

cos (\frac{π}{4}) cos (θ) + sin (\frac{π}{4}) sin (θ)

Simplify

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

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

= \frac{2}{2}

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

Simplify

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Use the following trivial identity:

sin (\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}

= \frac{2}{2}

= \frac{2}{2} cos (θ) + \frac{2}{2} sin (θ)

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

Multiply

\frac{2}{2} cos (θ) : \frac{2 cos ( θ )}{2}

\frac{2}{2} cos (θ)

Multiply fractions:

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

= \frac{2 cos ( θ )}{2}

= \frac{\frac{2}{2} cos ( θ ) - \frac{2}{2} sin ( θ )}{\frac{2 c o s ( θ )}{2} + \frac{2}{2} sin ( θ )}

Multiply

\frac{2}{2} sin (θ) : \frac{2 sin ( θ )}{2}

\frac{2}{2} sin (θ)

Multiply fractions:

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

= \frac{2 sin ( θ )}{2}

= \frac{\frac{2}{2} cos ( θ ) - \frac{2}{2} sin ( θ )}{\frac{2 c o s ( θ )}{2} + \frac{2 s i n ( θ )}{2}}

Multiply

\frac{2}{2} cos (θ) : \frac{2 cos ( θ )}{2}

\frac{2}{2} cos (θ)

Multiply fractions:

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

= \frac{2 cos ( θ )}{2}

= \frac{\frac{2 c o s ( θ )}{2} - \frac{2}{2} sin ( θ )}{\frac{2 c o s ( θ )}{2} + \frac{2 s i n ( θ )}{2}}

Multiply

\frac{2}{2} sin (θ) : \frac{2 sin ( θ )}{2}

\frac{2}{2} sin (θ)

Multiply fractions:

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

= \frac{2 sin ( θ )}{2}

= \frac{\frac{2 c o s ( θ )}{2} - \frac{2 s i n ( θ )}{2}}{\frac{2 c o s ( θ )}{2} + \frac{2 s i n ( θ )}{2}}

Combine the fractions

\frac{2 cos ( θ )}{2} + \frac{2 sin ( θ )}{2} : \frac{2 cos ( θ ) + 2 sin ( θ )}{2}

Apply rule

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

= \frac{2 cos ( θ ) + 2 sin ( θ )}{2}

= \frac{\frac{2 c o s ( θ )}{2} - \frac{2 s i n ( θ )}{2}}{\frac{2 c o s ( θ ) + 2 s i n ( θ )}{2}}

Combine the fractions

\frac{2 cos ( θ )}{2} - \frac{2 sin ( θ )}{2} : \frac{2 cos ( θ ) - 2 sin ( θ )}{2}

Apply rule

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

= \frac{2 cos ( θ ) - 2 sin ( θ )}{2}

= \frac{\frac{2 c o s ( θ ) - 2 s i n ( θ )}{2}}{\frac{2 c o s ( θ ) + 2 s i n ( θ )}{2}}

Divide fractions:

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

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

Cancel the common factor:

2

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

Factor out common term

2

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

Factor out common term

2

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

Cancel the common factor:

2

= \frac{cos ( θ ) - sin ( θ )}{cos ( θ ) + sin ( θ )}

= \frac{cos ( θ ) - sin ( θ )}{cos ( θ ) + sin ( θ )}

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

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

Subtract

\frac{cos ( θ ) - sin ( θ )}{cos ( θ ) + sin ( θ )}

from both sides

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

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

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

Rewrite using trig identities

- cos (θ) + sin (θ) + 1

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

sin (θ) - cos (θ)

Rewrite as

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

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 (θ) - sin (\frac{π}{4}) cos (θ))

Use the Angle Sum identity:

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

= 2 sin (θ - \frac{π}{4})

= 1 + 2 sin (θ - \frac{π}{4})

1 + 2 sin (θ - \frac{π}{4}) = 0

Move

1

to the right side

1 + 2 sin (θ - \frac{π}{4}) = 0

Subtract

1

from both sides

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

Simplify

2 sin (θ - \frac{π}{4}) = - 1

2 sin (θ - \frac{π}{4}) = - 1

Divide both sides by

2

2 sin (θ - \frac{π}{4}) = - 1

Divide both sides by

2

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

Simplify

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

Simplify

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

\frac{2 sin ( θ - \frac{π}{4} )}{2}

Cancel the common factor:

2

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

sin (θ - \frac{π}{4}) = - \frac{2}{2}

sin (θ - \frac{π}{4}) = - \frac{2}{2}

General solutions for

sin (θ - \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}

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

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

Solve

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

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

Move

\frac{π}{4}

to the right side

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

Add

\frac{π}{4}

to both sides

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

Simplify

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

Simplify

θ - \frac{π}{4} + \frac{π}{4} : θ

θ - \frac{π}{4} + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} = 0

= θ

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}

θ = 2 πn + \frac{3 π}{2}

θ = 2 πn + \frac{3 π}{2}

θ = 2 πn + \frac{3 π}{2}

Solve

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

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

Move

\frac{π}{4}

to the right side

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

Add

\frac{π}{4}

to both sides

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

Simplify

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

Simplify

θ - \frac{π}{4} + \frac{π}{4} : θ

θ - \frac{π}{4} + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} = 0

= θ

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

θ = 2 πn + 2 π

θ = 2 πn + 2 π

θ = 2 πn + 2 π

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

Since the equation is undefined for:

2 πn + \frac{3 π}{2}

θ = 2 πn + 2 π

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Frequently Asked Questions (FAQ)

What is the general solution for (cos(θ))/(1+sin(θ))=tan(pi/4-θ) ?
The general solution for (cos(θ))/(1+sin(θ))=tan(pi/4-θ) is θ=2pin+2pi