What is the general solution for tan(45)=(1-cos(u))/(sin(u)) ?

The general solution for tan(45)=(1-cos(u))/(sin(u)) is u=360n+90

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tan(45)=(1-cos(u))/(sin(u))

Solution

tan (4 5^{\circ}) = \frac{1 - cos ( u )}{sin ( u )}

Solution

u = 36 0^{\circ} n + 9 0^{\circ}

Radians

u = \frac{π}{2} + 2 πn

Solution steps

tan (4 5^{\circ}) = \frac{1 - cos ( u )}{sin ( u )}

tan (4 5^{\circ}) = 1

tan (4 5^{\circ})

Use the following trivial identity:

tan (4 5^{\circ}) = 1

tan (4 5^{\circ})

tan (x)

periodicity table with

18 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= 1

= 1

1 = \frac{1 - cos ( u )}{sin ( u )}

Subtract

\frac{1 - cos ( u )}{sin ( u )}

from both sides

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

Simplify

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

1 - \frac{1 - cos ( u )}{sin ( u )}

Convert element to fraction:

1 = \frac{1 sin ( u )}{sin ( u )}

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot sin (u) = sin (u)

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

- (1 - cos (u)) : - 1 + cos (u)

- (1 - cos (u))

Distribute parentheses

= - (1) - (- cos (u))

Apply minus-plus rules

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

= - 1 + cos (u)

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

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

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

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

Rewrite using trig identities

sin (u) - 1 + cos (u)

sin (u) + cos (u) = 2 sin (u + 4 5^{\circ})

sin (u) + cos (u)

Rewrite as

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

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 (4 5^{\circ}) sin (u) + sin (4 5^{\circ}) cos (u))

Use the Angle Sum identity:

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

= 2 sin (u + 4 5^{\circ})

= - 1 + 2 sin (u + 4 5^{\circ})

- 1 + 2 sin (u + 4 5^{\circ}) = 0

Move

1

to the right side

- 1 + 2 sin (u + 4 5^{\circ}) = 0

Add

1

to both sides

- 1 + 2 sin (u + 4 5^{\circ}) + 1 = 0 + 1

Simplify

2 sin (u + 4 5^{\circ}) = 1

2 sin (u + 4 5^{\circ}) = 1

Divide both sides by

2

2 sin (u + 4 5^{\circ}) = 1

Divide both sides by

2

\frac{2 sin ( u + 4 5 ^{\circ} )}{2} = \frac{1}{2}

Simplify

\frac{2 sin ( u + 4 5 ^{\circ} )}{2} = \frac{1}{2}

Simplify

\frac{2 sin ( u + 4 5 ^{\circ} )}{2} : sin (u + 4 5^{\circ})

\frac{2 sin ( u + 4 5 ^{\circ} )}{2}

Cancel the common factor:

2

= sin (u + 4 5^{\circ})

Simplify

\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}

sin (u + 4 5^{\circ}) = \frac{2}{2}

sin (u + 4 5^{\circ}) = \frac{2}{2}

sin (u + 4 5^{\circ}) = \frac{2}{2}

General solutions for

sin (u + 4 5^{\circ}) = \frac{2}{2}

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

u + 4 5^{\circ} = 4 5^{\circ} + 36 0^{\circ} n, u + 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n

u + 4 5^{\circ} = 4 5^{\circ} + 36 0^{\circ} n, u + 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n

Solve

u + 4 5^{\circ} = 4 5^{\circ} + 36 0^{\circ} n : u = 36 0^{\circ} n

u + 4 5^{\circ} = 4 5^{\circ} + 36 0^{\circ} n

Subtract

4 5^{\circ}

from both sides

u + 4 5^{\circ} - 4 5^{\circ} = 4 5^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Simplify

u = 36 0^{\circ} n

Solve

u + 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n : u = 36 0^{\circ} n + 9 0^{\circ}

u + 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n

Move

4 5^{\circ}

to the right side

u + 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n

Subtract

4 5^{\circ}

from both sides

u + 4 5^{\circ} - 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Simplify

u + 4 5^{\circ} - 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Simplify

u + 4 5^{\circ} - 4 5^{\circ} : u

u + 4 5^{\circ} - 4 5^{\circ}

Add similar elements:

4 5^{\circ} - 4 5^{\circ} = 0

= u

Simplify

13 5^{\circ} + 36 0^{\circ} n - 4 5^{\circ} : 36 0^{\circ} n + 9 0^{\circ}

13 5^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Group like terms

= 36 0^{\circ} n - 4 5^{\circ} + 13 5^{\circ}

Combine the fractions

- 4 5^{\circ} + 13 5^{\circ} : 9 0^{\circ}

Apply rule

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

= \frac{- 18 0 ^{\circ} + 54 0 ^{\circ}}{4}

Add similar elements:

- 18 0^{\circ} + 54 0^{\circ} = 36 0^{\circ}

= 9 0^{\circ}

Cancel the common factor:

2

= 9 0^{\circ}

= 36 0^{\circ} n + 9 0^{\circ}

u = 36 0^{\circ} n + 9 0^{\circ}

u = 36 0^{\circ} n + 9 0^{\circ}

u = 36 0^{\circ} n + 9 0^{\circ}

u = 36 0^{\circ} n, u = 36 0^{\circ} n + 9 0^{\circ}

Since the equation is undefined for:

36 0^{\circ} n

u = 36 0^{\circ} n + 9 0^{\circ}

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Frequently Asked Questions (FAQ)

What is the general solution for tan(45)=(1-cos(u))/(sin(u)) ?
The general solution for tan(45)=(1-cos(u))/(sin(u)) is u=360n+90