What is the general solution for cos(2x-180)-sin(x-90)=0 ?

The general solution for cos(2x-180)-sin(x-90)=0 is x=(720n)/3 ,x=120+(720n)/3

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cos(2x-180)-sin(x-90)=0

Solution

cos (2 x - 18 0^{\circ}) - sin (x - 9 0^{\circ}) = 0

Solution

x = \frac{72 0 ^{\circ} n}{3}, x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

Radians

x = 0 + \frac{4 π}{3} n, x = \frac{2 π}{3} + \frac{4 π}{3} n

Solution steps

cos (2 x - 18 0^{\circ}) - sin (x - 9 0^{\circ}) = 0

Rewrite using trig identities

cos (2 x - 18 0^{\circ}) - sin (x - 9 0^{\circ}) = 0

Rewrite using trig identities

sin (x - 9 0^{\circ})

Use the Angle Difference identity:

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

= sin (x) cos (9 0^{\circ}) - cos (x) sin (9 0^{\circ})

Simplify

sin (x) cos (9 0^{\circ}) - cos (x) sin (9 0^{\circ}) : - cos (x)

sin (x) cos (9 0^{\circ}) - cos (x) sin (9 0^{\circ})

sin (x) cos (9 0^{\circ}) = 0

sin (x) cos (9 0^{\circ})

Simplify

cos (9 0^{\circ}) : 0

cos (9 0^{\circ})

Use the following trivial identity:

cos (9 0^{\circ}) = 0

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

= 0

= 0 \cdot sin (x)

Apply rule

0 \cdot a = 0

= 0

cos (x) sin (9 0^{\circ}) = cos (x)

cos (x) sin (9 0^{\circ})

Simplify

sin (9 0^{\circ}) : 1

sin (9 0^{\circ})

Use the following trivial identity:

sin (9 0^{\circ}) = 1

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}

= 1

= 1 \cdot cos (x)

Multiply:

cos (x) \cdot 1 = cos (x)

= cos (x)

= 0 - cos (x)

0 - cos (x) = - cos (x)

= - cos (x)

= - cos (x)

Use the Angle Difference identity:

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

= cos (2 x) cos (18 0^{\circ}) + sin (2 x) sin (18 0^{\circ})

Simplify

cos (2 x) cos (18 0^{\circ}) + sin (2 x) sin (18 0^{\circ}) : - cos (2 x)

cos (2 x) cos (18 0^{\circ}) + sin (2 x) sin (18 0^{\circ})

cos (2 x) cos (18 0^{\circ}) = - cos (2 x)

cos (2 x) cos (18 0^{\circ})

Simplify

cos (18 0^{\circ}) : - 1

cos (18 0^{\circ})

Use the following trivial identity:

cos (18 0^{\circ}) = (- 1)

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

= - 1

= (- 1) cos (2 x)

Refine

= - cos (2 x)

= - cos (2 x) + sin (18 0^{\circ}) sin (2 x)

sin (2 x) sin (18 0^{\circ}) = 0

sin (2 x) sin (18 0^{\circ})

Simplify

sin (18 0^{\circ}) : 0

sin (18 0^{\circ})

Use the following trivial identity:

sin (18 0^{\circ}) = 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}

= 0

= 0 \cdot sin (2 x)

Apply rule

0 \cdot a = 0

= 0

= - cos (2 x) + 0

- cos (2 x) + 0 = - cos (2 x)

= - cos (2 x)

= - cos (2 x)

- cos (2 x) - (- cos (x)) = 0

Simplify

- cos (2 x) - (- cos (x)) : - cos (2 x) + cos (x)

- cos (2 x) - (- cos (x))

Apply rule

- (- a) = a

= - cos (2 x) + cos (x)

- cos (2 x) + cos (x) = 0

Use the Sum to Product identity:

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

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

Simplify

- 2 sin (\frac{x + 2 x}{2}) sin (\frac{x - 2 x}{2}) : 2 sin (\frac{x}{2}) sin (\frac{3 x}{2})

- 2 sin (\frac{x + 2 x}{2}) sin (\frac{x - 2 x}{2})

Add similar elements:

x + 2 x = 3 x

= - 2 sin (\frac{3 x}{2}) sin (\frac{x - 2 x}{2})

\frac{x - 2 x}{2} = - \frac{x}{2}

\frac{x - 2 x}{2}

Add similar elements:

x - 2 x = - x

= \frac{- x}{2}

Apply the fraction rule:

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

= - \frac{x}{2}

= - 2 sin (\frac{3 x}{2}) sin (- \frac{x}{2})

Use the negative angle identity:

sin (- x) = - sin (x)

= - 2 (- sin (\frac{x}{2})) sin (\frac{3 x}{2})

Apply rule

- (- a) = a

= 2 sin (\frac{x}{2}) sin (\frac{3 x}{2})

= 2 sin (\frac{x}{2}) sin (\frac{3 x}{2})

2 sin (\frac{3 x}{2}) sin (\frac{x}{2}) = 0

Solving each part separately

sin (\frac{3 x}{2}) = 0 or sin (\frac{x}{2}) = 0

sin (\frac{3 x}{2}) = 0 : x = \frac{72 0 ^{\circ} n}{3}, x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

sin (\frac{3 x}{2}) = 0

General solutions for

sin (\frac{3 x}{2}) = 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}

\frac{3 x}{2} = 0 + 36 0^{\circ} n, \frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

\frac{3 x}{2} = 0 + 36 0^{\circ} n, \frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

Solve

\frac{3 x}{2} = 0 + 36 0^{\circ} n : x = \frac{72 0 ^{\circ} n}{3}

\frac{3 x}{2} = 0 + 36 0^{\circ} n

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

\frac{3 x}{2} = 36 0^{\circ} n

Multiply both sides by

2

\frac{3 x}{2} = 36 0^{\circ} n

Multiply both sides by

2

\frac{2 \cdot 3 x}{2} = 2 \cdot 36 0^{\circ} n

Simplify

3 x = 72 0^{\circ} n

3 x = 72 0^{\circ} n

Divide both sides by

3

3 x = 72 0^{\circ} n

Divide both sides by

3

\frac{3 x}{3} = \frac{72 0 ^{\circ} n}{3}

Simplify

x = \frac{72 0 ^{\circ} n}{3}

x = \frac{72 0 ^{\circ} n}{3}

Solve

\frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n : x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

\frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

2

\frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

2

\frac{2 \cdot 3 x}{2} = 36 0^{\circ} + 2 \cdot 36 0^{\circ} n

Simplify

3 x = 36 0^{\circ} + 72 0^{\circ} n

3 x = 36 0^{\circ} + 72 0^{\circ} n

Divide both sides by

3

3 x = 36 0^{\circ} + 72 0^{\circ} n

Divide both sides by

3

\frac{3 x}{3} = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

Simplify

x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

x = \frac{72 0 ^{\circ} n}{3}, x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

sin (\frac{x}{2}) = 0 : x = 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n

sin (\frac{x}{2}) = 0

General solutions for

sin (\frac{x}{2}) = 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}

\frac{x}{2} = 0 + 36 0^{\circ} n, \frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n

\frac{x}{2} = 0 + 36 0^{\circ} n, \frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n

Solve

\frac{x}{2} = 0 + 36 0^{\circ} n : x = 72 0^{\circ} n

\frac{x}{2} = 0 + 36 0^{\circ} n

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

\frac{x}{2} = 36 0^{\circ} n

Multiply both sides by

2

\frac{x}{2} = 36 0^{\circ} n

Multiply both sides by

2

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

Simplify

x = 72 0^{\circ} n

x = 72 0^{\circ} n

Solve

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

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

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

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

Combine all the solutions

x = \frac{72 0 ^{\circ} n}{3}, x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}, x = 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n

Merge Overlapping Intervals

x = \frac{72 0 ^{\circ} n}{3}, x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2x-180)-sin(x-90)=0 ?
The general solution for cos(2x-180)-sin(x-90)=0 is x=(720n)/3 ,x=120+(720n)/3