What is the general solution for sin(x+20)+sin(40-x)=1 ?

The general solution for sin(x+20)+sin(40-x)=1 is x=360n+10

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sin(x+20)+sin(40-x)=1

Solution

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

Solution

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

Radians

x = \frac{π}{18} + 2 πn

Solution steps

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

Rewrite using trig identities

sin (x + 2 0^{\circ}) + sin (4 0^{\circ} - x)

Use the Sum to Product identity:

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

= 2 sin (\frac{x + 2 0 ^{\circ} + 4 0 ^{\circ} - x}{2}) cos (\frac{x + 2 0 ^{\circ} - ( 4 0 ^{\circ} - x )}{2})

Simplify

2 sin (\frac{x + 2 0 ^{\circ} + 4 0 ^{\circ} - x}{2}) cos (\frac{x + 2 0 ^{\circ} - ( 4 0 ^{\circ} - x )}{2}) : cos (\frac{18 x - 18 0 ^{\circ}}{18})

2 sin (\frac{x + 2 0 ^{\circ} + 4 0 ^{\circ} - x}{2}) cos (\frac{x + 2 0 ^{\circ} - ( 4 0 ^{\circ} - x )}{2})

\frac{x + 2 0 ^{\circ} + 4 0 ^{\circ} - x}{2} = 3 0^{\circ}

\frac{x + 2 0 ^{\circ} + 4 0 ^{\circ} - x}{2}

Combine the fractions

2 0^{\circ} + 4 0^{\circ} : 6 0^{\circ}

Apply rule

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

= \frac{18 0 ^{\circ} + 36 0 ^{\circ}}{9}

Add similar elements:

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

= 6 0^{\circ}

Cancel the common factor:

3

= 6 0^{\circ}

= \frac{x + 6 0 ^{\circ} - x}{2}

x + 6 0^{\circ} - x = 6 0^{\circ}

x + 6 0^{\circ} - x

Group like terms

= x - x + 6 0^{\circ}

Add similar elements:

x - x = 0

= 6 0^{\circ}

= \frac{6 0 ^{\circ}}{2}

Apply the fraction rule:

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

= \frac{18 0 ^{\circ}}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= 3 0^{\circ}

= 2 sin (3 0^{\circ}) cos (\frac{x - ( - x + 4 0 ^{\circ} ) + 2 0 ^{\circ}}{2})

\frac{x + 2 0 ^{\circ} - ( 4 0 ^{\circ} - x )}{2} = \frac{18 x - 18 0 ^{\circ}}{18}

\frac{x + 2 0 ^{\circ} - ( 4 0 ^{\circ} - x )}{2}

Join

x + 2 0^{\circ} - (4 0^{\circ} - x) : \frac{18 x - 18 0 ^{\circ}}{9}

x + 2 0^{\circ} - (4 0^{\circ} - x)

Convert element to fraction:

x = \frac{x 9}{9}, (- x + 4 0^{\circ}) = \frac{( 4 0 ^{\circ} - x ) 9}{9}

= \frac{x \cdot 9}{9} + 2 0^{\circ} - \frac{( 4 0 ^{\circ} - x ) \cdot 9}{9}

Since the denominators are equal, combine the fractions:

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

= \frac{x \cdot 9 + 18 0 ^{\circ} - ( 4 0 ^{\circ} - x ) \cdot 9}{9}

Expand

x \cdot 9 + 18 0^{\circ} - (4 0^{\circ} - x) \cdot 9 : 18 x - 18 0^{\circ}

x \cdot 9 + 18 0^{\circ} - (4 0^{\circ} - x) \cdot 9

= 9 x + 18 0^{\circ} - 9 (4 0^{\circ} - x)

Expand

- 9 (4 0^{\circ} - x) : - 36 0^{\circ} + 9 x

- 9 (4 0^{\circ} - x)

Apply the distributive law:

a (b - c) = ab - a c

a = - 9, b = 4 0^{\circ}, c = x

= - 9 \cdot 4 0^{\circ} - (- 9) x

Apply minus-plus rules

- (- a) = a

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

9 \cdot 4 0^{\circ} = 36 0^{\circ}

9 \cdot 4 0^{\circ}

Multiply fractions:

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

= 36 0^{\circ}

Cancel the common factor:

9

= 36 0^{\circ}

= - 36 0^{\circ} + 9 x

= x \cdot 9 + 18 0^{\circ} - 36 0^{\circ} + 9 x

Simplify

x \cdot 9 + 18 0^{\circ} - 36 0^{\circ} + 9 x : 18 x - 18 0^{\circ}

x \cdot 9 + 18 0^{\circ} - 36 0^{\circ} + 9 x

Group like terms

= 9 x + 9 x + 18 0^{\circ} - 36 0^{\circ}

Add similar elements:

9 x + 9 x = 18 x

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

Add similar elements:

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

= 18 x - 18 0^{\circ}

= 18 x - 18 0^{\circ}

= \frac{18 x - 18 0 ^{\circ}}{9}

= \frac{\frac{18 x - 18 0 ^{\circ}}{9}}{2}

Apply the fraction rule:

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

= \frac{18 x - 18 0 ^{\circ}}{9 \cdot 2}

Multiply the numbers:

9 \cdot 2 = 18

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

= 2 sin (3 0^{\circ}) cos (\frac{18 x - 18 0 ^{\circ}}{18})

Simplify

sin (3 0^{\circ}) : \frac{1}{2}

sin (3 0^{\circ})

Use the following trivial identity:

sin (3 0^{\circ}) = \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{1}{2}

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

Multiply:

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

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

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

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

General solutions for

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

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

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

Solve

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

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

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

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

Multiply both sides by

18

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

Multiply both sides by

18

\frac{18 ( 18 x - 18 0 ^{\circ} )}{18} = 18 \cdot 36 0^{\circ} n

Simplify

18 x - 18 0^{\circ} = 648 0^{\circ} n

18 x - 18 0^{\circ} = 648 0^{\circ} n

Move

18 0^{\circ}

to the right side

18 x - 18 0^{\circ} = 648 0^{\circ} n

Add

18 0^{\circ}

to both sides

18 x - 18 0^{\circ} + 18 0^{\circ} = 648 0^{\circ} n + 18 0^{\circ}

Simplify

18 x = 648 0^{\circ} n + 18 0^{\circ}

18 x = 648 0^{\circ} n + 18 0^{\circ}

Divide both sides by

18

18 x = 648 0^{\circ} n + 18 0^{\circ}

Divide both sides by

18

\frac{18 x}{18} = \frac{648 0 ^{\circ} n}{18} + 1 0^{\circ}

Simplify

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

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

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

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x+20)+sin(40-x)=1 ?
The general solution for sin(x+20)+sin(40-x)=1 is x=360n+10