What is the general solution for sin(2x+110)+sin(2x-10)= 1/2 ?

The general solution for sin(2x+110)+sin(2x-10)= 1/2 is x=-10+180n,x=50+180n

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sin(2x+110)+sin(2x-10)= 1/2

Solution

sin (2 x + 11 0^{\circ}) + sin (2 x - 1 0^{\circ}) = \frac{1}{2}

Solution

x = - 1 0^{\circ} + 18 0^{\circ} n, x = 5 0^{\circ} + 18 0^{\circ} n

Radians

x = - \frac{π}{18} + πn, x = \frac{5 π}{18} + πn

Solution steps

sin (2 x + 11 0^{\circ}) + sin (2 x - 1 0^{\circ}) = \frac{1}{2}

Rewrite using trig identities

sin (2 x + 11 0^{\circ}) + sin (2 x - 1 0^{\circ})

Use the Sum to Product identity:

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

= 2 sin (\frac{2 x + 11 0 ^{\circ} + 2 x - 1 0 ^{\circ}}{2}) cos (\frac{2 x + 11 0 ^{\circ} - ( 2 x - 1 0 ^{\circ} )}{2})

Simplify

2 sin (\frac{2 x + 11 0 ^{\circ} + 2 x - 1 0 ^{\circ}}{2}) cos (\frac{2 x + 11 0 ^{\circ} - ( 2 x - 1 0 ^{\circ} )}{2}) : sin (\frac{36 x + 90 0 ^{\circ}}{18})

2 sin (\frac{2 x + 11 0 ^{\circ} + 2 x - 1 0 ^{\circ}}{2}) cos (\frac{2 x + 11 0 ^{\circ} - ( 2 x - 1 0 ^{\circ} )}{2})

\frac{2 x + 11 0 ^{\circ} + 2 x - 1 0 ^{\circ}}{2} = \frac{36 x + 90 0 ^{\circ}}{18}

\frac{2 x + 11 0 ^{\circ} + 2 x - 1 0 ^{\circ}}{2}

Combine the fractions

11 0^{\circ} - 1 0^{\circ} : 10 0^{\circ}

Apply rule

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

= \frac{198 0 ^{\circ} - 18 0 ^{\circ}}{18}

Add similar elements:

198 0^{\circ} - 18 0^{\circ} = 180 0^{\circ}

= 10 0^{\circ}

Cancel the common factor:

2

= 10 0^{\circ}

= \frac{2 x + 10 0 ^{\circ} + 2 x}{2}

2 x + 10 0^{\circ} + 2 x = 4 x + 10 0^{\circ}

2 x + 10 0^{\circ} + 2 x

Group like terms

= 2 x + 2 x + 10 0^{\circ}

Add similar elements:

2 x + 2 x = 4 x

= 4 x + 10 0^{\circ}

= \frac{4 x + 10 0 ^{\circ}}{2}

Join

4 x + 10 0^{\circ} : \frac{36 x + 90 0 ^{\circ}}{9}

4 x + 10 0^{\circ}

Convert element to fraction:

4 x = \frac{4 x 9}{9}

= \frac{4 x \cdot 9}{9} + 10 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{4 x \cdot 9 + 90 0 ^{\circ}}{9}

Multiply the numbers:

4 \cdot 9 = 36

= \frac{36 x + 90 0 ^{\circ}}{9}

= \frac{\frac{36 x + 90 0 ^{\circ}}{9}}{2}

Apply the fraction rule:

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

= \frac{36 x + 90 0 ^{\circ}}{9 \cdot 2}

Multiply the numbers:

9 \cdot 2 = 18

= \frac{36 x + 90 0 ^{\circ}}{18}

= 2 sin (\frac{36 x + 90 0 ^{\circ}}{18}) cos (\frac{2 x - ( 2 x - 1 0 ^{\circ} ) + 11 0 ^{\circ}}{2})

\frac{2 x + 11 0 ^{\circ} - ( 2 x - 1 0 ^{\circ} )}{2} = 6 0^{\circ}

\frac{2 x + 11 0 ^{\circ} - ( 2 x - 1 0 ^{\circ} )}{2}

Join

2 x + 11 0^{\circ} - (2 x - 1 0^{\circ}) : 12 0^{\circ}

2 x + 11 0^{\circ} - (2 x - 1 0^{\circ})

Convert element to fraction:

2 x = \frac{2 x 18}{18}, (2 x - 1 0^{\circ}) = \frac{( 2 x - 1 0 ^{\circ} ) 18}{18}

= \frac{2 x \cdot 18}{18} + 11 0^{\circ} - \frac{( 2 x - 1 0 ^{\circ} ) \cdot 18}{18}

Since the denominators are equal, combine the fractions:

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

= \frac{2 x \cdot 18 + 198 0 ^{\circ} - ( 2 x - 1 0 ^{\circ} ) \cdot 18}{18}

Multiply the numbers:

2 \cdot 18 = 36

= \frac{36 x + 198 0 ^{\circ} - 18 ( 2 x - 1 0 ^{\circ} )}{18}

Expand

36 x + 198 0^{\circ} - (2 x - 1 0^{\circ}) \cdot 18 : 216 0^{\circ}

36 x + 198 0^{\circ} - (2 x - 1 0^{\circ}) \cdot 18

= 36 x + 198 0^{\circ} - 18 (2 x - 1 0^{\circ})

Expand

- 18 (2 x - 1 0^{\circ}) : - 36 x + 18 0^{\circ}

- 18 (2 x - 1 0^{\circ})

Apply the distributive law:

a (b - c) = ab - a c

a = - 18, b = 2 x, c = 1 0^{\circ}

= - 18 \cdot 2 x - (- 18) 1 0^{\circ}

Apply minus-plus rules

- (- a) = a

= - 18 \cdot 2 x + 18 \cdot 1 0^{\circ}

Simplify

- 18 \cdot 2 x + 18 \cdot 1 0^{\circ} : - 36 x + 18 0^{\circ}

- 18 \cdot 2 x + 18 \cdot 1 0^{\circ}

18 \cdot 2 x = 36 x

18 \cdot 2 x

Multiply the numbers:

18 \cdot 2 = 36

= 36 x

18 \cdot 1 0^{\circ} = 18 0^{\circ}

18 \cdot 1 0^{\circ}

Multiply fractions:

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

= 18 0^{\circ}

Cancel the common factor:

18

= 18 0^{\circ}

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

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

= 36 x + 198 0^{\circ} - 36 x + 18 0^{\circ}

Simplify

36 x + 198 0^{\circ} - 36 x + 18 0^{\circ} : 216 0^{\circ}

36 x + 198 0^{\circ} - 36 x + 18 0^{\circ}

Group like terms

= 36 x - 36 x + 198 0^{\circ} + 18 0^{\circ}

Add similar elements:

36 x - 36 x = 0

= 198 0^{\circ} + 18 0^{\circ}

Add similar elements:

198 0^{\circ} + 18 0^{\circ} = 216 0^{\circ}

= 216 0^{\circ}

= 216 0^{\circ}

= 12 0^{\circ}

Cancel the common factor:

6

= 12 0^{\circ}

= \frac{12 0 ^{\circ}}{2}

Apply the fraction rule:

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

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

Multiply the numbers:

3 \cdot 2 = 6

= 6 0^{\circ}

Cancel the common factor:

2

= 6 0^{\circ}

= 2 cos (6 0^{\circ}) sin (\frac{36 x + 90 0 ^{\circ}}{18})

Simplify

cos (6 0^{\circ}) : \frac{1}{2}

cos (6 0^{\circ})

Use the following trivial identity:

cos (6 0^{\circ}) = \frac{1}{2}

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}

= \frac{1}{2}

= 2 \cdot \frac{1}{2} sin (\frac{36 x + 90 0 ^{\circ}}{18})

Multiply fractions:

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

= \frac{1 \cdot 2}{2} sin (\frac{36 x + 90 0 ^{\circ}}{18})

Cancel the common factor:

2

= sin (\frac{36 x + 90 0 ^{\circ}}{18}) \cdot 1

Multiply:

sin (\frac{36 x + 90 0 ^{\circ}}{18}) \cdot 1 = sin (\frac{36 x + 90 0 ^{\circ}}{18})

= sin (\frac{36 x + 90 0 ^{\circ}}{18})

= sin (\frac{36 x + 90 0 ^{\circ}}{18})

sin (\frac{36 x + 90 0 ^{\circ}}{18}) = \frac{1}{2}

General solutions for

sin (\frac{36 x + 90 0 ^{\circ}}{18}) = \frac{1}{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}

\frac{36 x + 90 0 ^{\circ}}{18} = 3 0^{\circ} + 36 0^{\circ} n, \frac{36 x + 90 0 ^{\circ}}{18} = 15 0^{\circ} + 36 0^{\circ} n

\frac{36 x + 90 0 ^{\circ}}{18} = 3 0^{\circ} + 36 0^{\circ} n, \frac{36 x + 90 0 ^{\circ}}{18} = 15 0^{\circ} + 36 0^{\circ} n

Solve

\frac{36 x + 90 0 ^{\circ}}{18} = 3 0^{\circ} + 36 0^{\circ} n : x = - 1 0^{\circ} + 18 0^{\circ} n

\frac{36 x + 90 0 ^{\circ}}{18} = 3 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

18

\frac{36 x + 90 0 ^{\circ}}{18} = 3 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

18

\frac{18 ( 36 x + 90 0 ^{\circ} )}{18} = 18 \cdot 3 0^{\circ} + 18 \cdot 36 0^{\circ} n

Simplify

\frac{18 ( 36 x + 90 0 ^{\circ} )}{18} = 18 \cdot 3 0^{\circ} + 18 \cdot 36 0^{\circ} n

Simplify

\frac{18 ( 36 x + 90 0 ^{\circ} )}{18} : 36 x + 90 0^{\circ}

\frac{18 ( 36 x + 90 0 ^{\circ} )}{18}

Divide the numbers:

\frac{18}{18} = 1

= 36 x + 90 0^{\circ}

Simplify

18 \cdot 3 0^{\circ} + 18 \cdot 36 0^{\circ} n : 54 0^{\circ} + 648 0^{\circ} n

18 \cdot 3 0^{\circ} + 18 \cdot 36 0^{\circ} n

18 \cdot 3 0^{\circ} = 54 0^{\circ}

18 \cdot 3 0^{\circ}

Multiply fractions:

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

= 54 0^{\circ}

Divide the numbers:

\frac{18}{6} = 3

= 54 0^{\circ}

18 \cdot 36 0^{\circ} n = 648 0^{\circ} n

18 \cdot 36 0^{\circ} n

Multiply the numbers:

18 \cdot 2 = 36

= 648 0^{\circ} n

= 54 0^{\circ} + 648 0^{\circ} n

36 x + 90 0^{\circ} = 54 0^{\circ} + 648 0^{\circ} n

36 x + 90 0^{\circ} = 54 0^{\circ} + 648 0^{\circ} n

36 x + 90 0^{\circ} = 54 0^{\circ} + 648 0^{\circ} n

Move

90 0^{\circ}

to the right side

36 x + 90 0^{\circ} = 54 0^{\circ} + 648 0^{\circ} n

Subtract

90 0^{\circ}

from both sides

36 x + 90 0^{\circ} - 90 0^{\circ} = 54 0^{\circ} + 648 0^{\circ} n - 90 0^{\circ}

Simplify

36 x = - 36 0^{\circ} + 648 0^{\circ} n

36 x = - 36 0^{\circ} + 648 0^{\circ} n

Divide both sides by

36

36 x = - 36 0^{\circ} + 648 0^{\circ} n

Divide both sides by

36

\frac{36 x}{36} = - 1 0^{\circ} + \frac{648 0 ^{\circ} n}{36}

Simplify

\frac{36 x}{36} = - 1 0^{\circ} + \frac{648 0 ^{\circ} n}{36}

Simplify

\frac{36 x}{36} : x

\frac{36 x}{36}

Divide the numbers:

\frac{36}{36} = 1

= x

Simplify

- 1 0^{\circ} + \frac{648 0 ^{\circ} n}{36} : - 1 0^{\circ} + 18 0^{\circ} n

- 1 0^{\circ} + \frac{648 0 ^{\circ} n}{36}

Cancel

1 0^{\circ} : 1 0^{\circ}

1 0^{\circ}

Cancel the common factor:

2

= 1 0^{\circ}

= - 1 0^{\circ} + \frac{648 0 ^{\circ} n}{36}

Divide the numbers:

\frac{36}{36} = 1

= - 1 0^{\circ} + 18 0^{\circ} n

x = - 1 0^{\circ} + 18 0^{\circ} n

x = - 1 0^{\circ} + 18 0^{\circ} n

x = - 1 0^{\circ} + 18 0^{\circ} n

Solve

\frac{36 x + 90 0 ^{\circ}}{18} = 15 0^{\circ} + 36 0^{\circ} n : x = 5 0^{\circ} + 18 0^{\circ} n

\frac{36 x + 90 0 ^{\circ}}{18} = 15 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

18

\frac{36 x + 90 0 ^{\circ}}{18} = 15 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

18

\frac{18 ( 36 x + 90 0 ^{\circ} )}{18} = 18 \cdot 15 0^{\circ} + 18 \cdot 36 0^{\circ} n

Simplify

\frac{18 ( 36 x + 90 0 ^{\circ} )}{18} = 18 \cdot 15 0^{\circ} + 18 \cdot 36 0^{\circ} n

Simplify

\frac{18 ( 36 x + 90 0 ^{\circ} )}{18} : 36 x + 90 0^{\circ}

\frac{18 ( 36 x + 90 0 ^{\circ} )}{18}

Divide the numbers:

\frac{18}{18} = 1

= 36 x + 90 0^{\circ}

Simplify

18 \cdot 15 0^{\circ} + 18 \cdot 36 0^{\circ} n : 270 0^{\circ} + 648 0^{\circ} n

18 \cdot 15 0^{\circ} + 18 \cdot 36 0^{\circ} n

18 \cdot 15 0^{\circ} = 270 0^{\circ}

18 \cdot 15 0^{\circ}

Multiply fractions:

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

= 270 0^{\circ}

Multiply the numbers:

5 \cdot 18 = 90

= 270 0^{\circ}

Divide the numbers:

\frac{90}{6} = 15

= 270 0^{\circ}

18 \cdot 36 0^{\circ} n = 648 0^{\circ} n

18 \cdot 36 0^{\circ} n

Multiply the numbers:

18 \cdot 2 = 36

= 648 0^{\circ} n

= 270 0^{\circ} + 648 0^{\circ} n

36 x + 90 0^{\circ} = 270 0^{\circ} + 648 0^{\circ} n

36 x + 90 0^{\circ} = 270 0^{\circ} + 648 0^{\circ} n

36 x + 90 0^{\circ} = 270 0^{\circ} + 648 0^{\circ} n

Move

90 0^{\circ}

to the right side

36 x + 90 0^{\circ} = 270 0^{\circ} + 648 0^{\circ} n

Subtract

90 0^{\circ}

from both sides

36 x + 90 0^{\circ} - 90 0^{\circ} = 270 0^{\circ} + 648 0^{\circ} n - 90 0^{\circ}

Simplify

36 x = 180 0^{\circ} + 648 0^{\circ} n

36 x = 180 0^{\circ} + 648 0^{\circ} n

Divide both sides by

36

36 x = 180 0^{\circ} + 648 0^{\circ} n

Divide both sides by

36

\frac{36 x}{36} = 5 0^{\circ} + \frac{648 0 ^{\circ} n}{36}

Simplify

\frac{36 x}{36} = 5 0^{\circ} + \frac{648 0 ^{\circ} n}{36}

Simplify

\frac{36 x}{36} : x

\frac{36 x}{36}

Divide the numbers:

\frac{36}{36} = 1

= x

Simplify

5 0^{\circ} + \frac{648 0 ^{\circ} n}{36} : 5 0^{\circ} + 18 0^{\circ} n

5 0^{\circ} + \frac{648 0 ^{\circ} n}{36}

Cancel

5 0^{\circ} : 5 0^{\circ}

5 0^{\circ}

Cancel the common factor:

2

= 5 0^{\circ}

= 5 0^{\circ} + \frac{648 0 ^{\circ} n}{36}

Divide the numbers:

\frac{36}{36} = 1

= 5 0^{\circ} + 18 0^{\circ} n

x = 5 0^{\circ} + 18 0^{\circ} n

x = 5 0^{\circ} + 18 0^{\circ} n

x = 5 0^{\circ} + 18 0^{\circ} n

x = - 1 0^{\circ} + 18 0^{\circ} n, x = 5 0^{\circ} + 18 0^{\circ} n

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(2x+110)+sin(2x-10)= 1/2 ?
The general solution for sin(2x+110)+sin(2x-10)= 1/2 is x=-10+180n,x=50+180n