What is the general solution for tan(-60)=((1-cos(u)))/(sin(u)) ?

The general solution for tan(-60)=((1-cos(u)))/(sin(u)) is u=720n,u=360+720n,u=240+360n

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tan(-60)=((1-cos(u)))/(sin(u))

Solution

tan (- 6 0^{\circ}) = \frac{( 1 - cos ( u ) )}{sin ( u )}

Solution

u = 72 0^{\circ} n, u = 36 0^{\circ} + 72 0^{\circ} n, u = 24 0^{\circ} + 36 0^{\circ} n

Radians

u = 0 + 4 πn, u = 2 π + 4 πn, u = \frac{4 π}{3} + 2 πn

Solution steps

tan (- 6 0^{\circ}) = \frac{( 1 - cos ( u ) )}{sin ( u )}

tan (- 6 0^{\circ}) = - 3

tan (- 6 0^{\circ})

Use the following property:

tan (- x) = - tan (x)

tan (- 6 0^{\circ}) = - tan (6 0^{\circ})

= - tan (6 0^{\circ})

Use the following trivial identity:

tan (6 0^{\circ}) = 3

tan (6 0^{\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}

= 3

= - 3

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

Subtract

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

from both sides

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

Simplify

- 3 - \frac{1 - cos ( u )}{sin ( u )} : \frac{- 3 sin ( u ) - 1 + cos ( u )}{sin ( u )}

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

Convert element to fraction:

3 = \frac{3 sin ( u )}{sin ( u )}

= - \frac{3 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{- 3 sin ( u ) - ( 1 - cos ( u ) )}{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{- 3 sin ( u ) - 1 + cos ( u )}{sin ( u )}

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

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

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

Let:

v = \frac{1}{2} u

- 3 sin (2 v) - 1 + cos (2 v) = 0

Rewrite using trig identities

- 1 + cos (2 v) - sin (2 v) 3

Use the Double Angle identity:

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

= - 1 + cos (2 v) - 3 \cdot 2 sin (v) cos (v)

Use the Double Angle identity:

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

= - 1 + 1 - 2 sin^{2} (v) - 23 cos (v) sin (v)

Simplify

= - 2 sin^{2} (v) - 23 cos (v) sin (v)

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

Factor

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

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

Apply exponent rule:

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

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

= - 2 sin (v) sin (v) - 2 cos (v) sin (v) 3

Rewrite as

= - 2 sin (v) sin (v) - 2 sin (v) cos (v) 3

Factor out common term

2 sin (v)

= - 2 sin (v) (sin (v) + cos (v) 3)

- 2 sin (v) (sin (v) + 3 cos (v)) = 0

Solving each part separately

sin (v) = 0 or sin (v) + 3 cos (v) = 0

sin (v) = 0 : v = 36 0^{\circ} n, v = 18 0^{\circ} + 36 0^{\circ} n

sin (v) = 0

General solutions for

sin (v) = 0

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}

v = 0 + 36 0^{\circ} n, v = 18 0^{\circ} + 36 0^{\circ} n

v = 0 + 36 0^{\circ} n, v = 18 0^{\circ} + 36 0^{\circ} n

Solve

v = 0 + 36 0^{\circ} n : v = 36 0^{\circ} n

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

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

v = 36 0^{\circ} n

v = 36 0^{\circ} n, v = 18 0^{\circ} + 36 0^{\circ} n

sin (v) + 3 cos (v) = 0 : v = 12 0^{\circ} + 18 0^{\circ} n

sin (v) + 3 cos (v) = 0

Rewrite using trig identities

sin (v) + 3 cos (v) = 0

Divide both sides by

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

\frac{sin ( v ) + 3 cos ( v )}{cos ( v )} = \frac{0}{cos ( v )}

Simplify

\frac{sin ( v )}{cos ( v )} + 3 = 0

Use the basic trigonometric identity:

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

tan (v) + 3 = 0

tan (v) + 3 = 0

Move

3

to the right side

tan (v) + 3 = 0

Subtract

3

from both sides

tan (v) + 3 - 3 = 0 - 3

Simplify

tan (v) = - 3

tan (v) = - 3

General solutions for

tan (v) = - 3

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}

v = 12 0^{\circ} + 18 0^{\circ} n

v = 12 0^{\circ} + 18 0^{\circ} n

Combine all the solutions

v = 36 0^{\circ} n, v = 18 0^{\circ} + 36 0^{\circ} n, v = 12 0^{\circ} + 18 0^{\circ} n

Substitute back

v = \frac{1}{2} u

\frac{1}{2} u = 36 0^{\circ} n : u = 72 0^{\circ} n

\frac{1}{2} u = 36 0^{\circ} n

Multiply both sides by

2

\frac{1}{2} u = 36 0^{\circ} n

Multiply both sides by

2

2 \cdot \frac{1}{2} u = 2 \cdot 36 0^{\circ} n

Simplify

u = 72 0^{\circ} n

u = 72 0^{\circ} n

\frac{1}{2} u = 18 0^{\circ} + 36 0^{\circ} n : u = 36 0^{\circ} + 72 0^{\circ} n

\frac{1}{2} u = 18 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

2

\frac{1}{2} u = 18 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

2

2 \cdot \frac{1}{2} u = 36 0^{\circ} + 2 \cdot 36 0^{\circ} n

Simplify

u = 36 0^{\circ} + 72 0^{\circ} n

u = 36 0^{\circ} + 72 0^{\circ} n

\frac{1}{2} u = 12 0^{\circ} + 18 0^{\circ} n : u = 24 0^{\circ} + 36 0^{\circ} n

\frac{1}{2} u = 12 0^{\circ} + 18 0^{\circ} n

Multiply both sides by

2

\frac{1}{2} u = 12 0^{\circ} + 18 0^{\circ} n

Multiply both sides by

2

2 \cdot \frac{1}{2} u = 2 \cdot 12 0^{\circ} + 36 0^{\circ} n

Simplify

2 \cdot \frac{1}{2} u = 2 \cdot 12 0^{\circ} + 36 0^{\circ} n

Simplify

2 \cdot \frac{1}{2} u : u

2 \cdot \frac{1}{2} u

Multiply fractions:

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

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

Cancel the common factor:

2

= u \cdot 1

Multiply:

u \cdot 1 = u

= u

Simplify

2 \cdot 12 0^{\circ} + 36 0^{\circ} n : 24 0^{\circ} + 36 0^{\circ} n

2 \cdot 12 0^{\circ} + 36 0^{\circ} n

2 \cdot 12 0^{\circ} = 24 0^{\circ}

2 \cdot 12 0^{\circ}

Multiply fractions:

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

= 24 0^{\circ}

Multiply the numbers:

2 \cdot 2 = 4

= 24 0^{\circ}

= 24 0^{\circ} + 36 0^{\circ} n

u = 24 0^{\circ} + 36 0^{\circ} n

u = 24 0^{\circ} + 36 0^{\circ} n

u = 24 0^{\circ} + 36 0^{\circ} n

u = 72 0^{\circ} n, u = 36 0^{\circ} + 72 0^{\circ} n, u = 24 0^{\circ} + 36 0^{\circ} n

Since the equation is undefined for:

72 0^{\circ} n, 36 0^{\circ} + 72 0^{\circ} n

u = 72 0^{\circ} n, u = 36 0^{\circ} + 72 0^{\circ} n, u = 24 0^{\circ} + 36 0^{\circ} n

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for tan(-60)=((1-cos(u)))/(sin(u)) ?
The general solution for tan(-60)=((1-cos(u)))/(sin(u)) is u=720n,u=360+720n,u=240+360n