What is the general solution for cos(θ)=1+sin(θ) ?

The general solution for cos(θ)=1+sin(θ) is θ=(3pi)/2+2pin,θ=2pi+2pin

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cos(θ)=1+sin(θ)

Solution

cos (θ) = 1 + sin (θ)

Solution

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

Degrees

θ = 27 0^{\circ} + 36 0^{\circ} n, θ = 36 0^{\circ} + 36 0^{\circ} n

Solution steps

cos (θ) = 1 + sin (θ)

Subtract

1 + sin (θ)

from both sides

cos (θ) - 1 - sin (θ) = 0

Rewrite using trig identities

- 1 + cos (θ) - sin (θ)

Use the following identity:

sin (x) = cos (\frac{π}{2} - x)

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

Use the Sum to Product identity:

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

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

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

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

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

\frac{θ + \frac{π}{2} - θ}{2}

θ + \frac{π}{2} - θ = \frac{π}{2}

θ + \frac{π}{2} - θ

Group like terms

= θ - θ + \frac{π}{2}

Add similar elements:

θ - θ = 0

= \frac{π}{2}

= \frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

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

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

\frac{θ - ( \frac{π}{2} - θ )}{2}

Expand

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

θ - (\frac{π}{2} - θ)

- (\frac{π}{2} - θ) : - \frac{π}{2} + θ

- (\frac{π}{2} - θ)

Distribute parentheses

= - (\frac{π}{2}) - (- θ)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - \frac{π}{2} + θ

= θ - \frac{π}{2} + θ

Simplify

θ - \frac{π}{2} + θ : 2 θ - \frac{π}{2}

θ - \frac{π}{2} + θ

Group like terms

= θ + θ - \frac{π}{2}

Add similar elements:

θ + θ = 2 θ

= 2 θ - \frac{π}{2}

= 2 θ - \frac{π}{2}

= \frac{2 θ - \frac{π}{2}}{2}

Join

2 θ - \frac{π}{2} : \frac{4 θ - π}{2}

2 θ - \frac{π}{2}

Convert element to fraction:

2 θ = \frac{2 θ 2}{2}

= \frac{2 θ \cdot 2}{2} - \frac{π}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{2 θ \cdot 2 - π}{2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 θ - π}{2}

= \frac{\frac{4 θ - π}{2}}{2}

Apply the fraction rule:

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

= \frac{4 θ - π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 θ - π}{4}

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

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}

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 2 sin (\frac{4 θ - π}{4})

= - 1 - 2 sin (\frac{4 θ - π}{4})

- 1 - 2 sin (\frac{4 θ - π}{4}) = 0

Move

1

to the right side

- 1 - 2 sin (\frac{4 θ - π}{4}) = 0

Add

1

to both sides

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

Simplify

- 2 sin (\frac{4 θ - π}{4}) = 1

- 2 sin (\frac{4 θ - π}{4}) = 1

Divide both sides by

- 2

- 2 sin (\frac{4 θ - π}{4}) = 1

Divide both sides by

- 2

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

Simplify

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

Simplify

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

2

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

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

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

General solutions for

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

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

Solve

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

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

Multiply both sides by

4

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

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 ( 4 θ - π )}{4} : 4 θ - π

\frac{4 ( 4 θ - π )}{4}

Divide the numbers:

\frac{4}{4} = 1

= 4 θ - π

Simplify

4 \cdot \frac{5 π}{4} + 4 \cdot 2 πn : 5 π + 8 πn

4 \cdot \frac{5 π}{4} + 4 \cdot 2 πn

4 \cdot \frac{5 π}{4} = 5 π

4 \cdot \frac{5 π}{4}

Multiply fractions:

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

= \frac{5 π 4}{4}

Cancel the common factor:

4

= 5 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= 5 π + 8 πn

4 θ - π = 5 π + 8 πn

4 θ - π = 5 π + 8 πn

4 θ - π = 5 π + 8 πn

Move

π

to the right side

4 θ - π = 5 π + 8 πn

Add

π

to both sides

4 θ - π + π = 5 π + 8 πn + π

Simplify

4 θ = 6 π + 8 πn

4 θ = 6 π + 8 πn

Divide both sides by

4

4 θ = 6 π + 8 πn

Divide both sides by

4

\frac{4 θ}{4} = \frac{6 π}{4} + \frac{8 πn}{4}

Simplify

\frac{4 θ}{4} = \frac{6 π}{4} + \frac{8 πn}{4}

Simplify

\frac{4 θ}{4} : θ

\frac{4 θ}{4}

Divide the numbers:

\frac{4}{4} = 1

= θ

Simplify

\frac{6 π}{4} + \frac{8 πn}{4} : \frac{3 π}{2} + 2 πn

\frac{6 π}{4} + \frac{8 πn}{4}

Cancel

\frac{6 π}{4} : \frac{3 π}{2}

\frac{6 π}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

= \frac{3 π}{2} + \frac{8 πn}{4}

Divide the numbers:

\frac{8}{4} = 2

= \frac{3 π}{2} + 2 πn

θ = \frac{3 π}{2} + 2 πn

θ = \frac{3 π}{2} + 2 πn

θ = \frac{3 π}{2} + 2 πn

Solve

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

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

Multiply both sides by

4

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

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 ( 4 θ - π )}{4} : 4 θ - π

\frac{4 ( 4 θ - π )}{4}

Divide the numbers:

\frac{4}{4} = 1

= 4 θ - π

Simplify

4 \cdot \frac{7 π}{4} + 4 \cdot 2 πn : 7 π + 8 πn

4 \cdot \frac{7 π}{4} + 4 \cdot 2 πn

4 \cdot \frac{7 π}{4} = 7 π

4 \cdot \frac{7 π}{4}

Multiply fractions:

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

= \frac{7 π 4}{4}

Cancel the common factor:

4

= 7 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= 7 π + 8 πn

4 θ - π = 7 π + 8 πn

4 θ - π = 7 π + 8 πn

4 θ - π = 7 π + 8 πn

Move

π

to the right side

4 θ - π = 7 π + 8 πn

Add

π

to both sides

4 θ - π + π = 7 π + 8 πn + π

Simplify

4 θ = 8 π + 8 πn

4 θ = 8 π + 8 πn

Divide both sides by

4

4 θ = 8 π + 8 πn

Divide both sides by

4

\frac{4 θ}{4} = \frac{8 π}{4} + \frac{8 πn}{4}

Simplify

θ = 2 π + 2 πn

θ = 2 π + 2 πn

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

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Frequently Asked Questions (FAQ)

What is the general solution for cos(θ)=1+sin(θ) ?
The general solution for cos(θ)=1+sin(θ) is θ=(3pi)/2+2pin,θ=2pi+2pin