What is the general solution for sin(x)+sin(90-x)=sin(90) ?

The general solution for sin(x)+sin(90-x)=sin(90) is x=90+360n,x=360+360n

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sin(x)+sin(90-x)=sin(90)

Solution

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

Solution

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

Radians

x = \frac{π}{2} + 2 πn, x = 2 π + 2 πn

Solution steps

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

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

sin (9 0^{\circ})

Use the following trivial identity:

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

sin (9 0^{\circ})

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

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

Rewrite using trig identities

sin (x) + sin (9 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 + 9 0 ^{\circ} - x}{2}) cos (\frac{x - ( 9 0 ^{\circ} - x )}{2})

Simplify

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

2 sin (\frac{x + 9 0 ^{\circ} - x}{2}) cos (\frac{x - ( 9 0 ^{\circ} - x )}{2})

\frac{x + 9 0 ^{\circ} - x}{2} = 4 5^{\circ}

\frac{x + 9 0 ^{\circ} - x}{2}

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

x + 9 0^{\circ} - x

Group like terms

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

Add similar elements:

x - x = 0

= 9 0^{\circ}

= \frac{9 0 ^{\circ}}{2}

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= 4 5^{\circ}

= 2 sin (4 5^{\circ}) cos (\frac{x - ( - x + 9 0 ^{\circ} )}{2})

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

\frac{x - ( 9 0 ^{\circ} - x )}{2}

Expand

x - (9 0^{\circ} - x) : 2 x - 9 0^{\circ}

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

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

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

Distribute parentheses

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

Apply minus-plus rules

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

= - 9 0^{\circ} + x

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

Simplify

x - 9 0^{\circ} + x : 2 x - 9 0^{\circ}

x - 9 0^{\circ} + x

Group like terms

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

Add similar elements:

x + x = 2 x

= 2 x - 9 0^{\circ}

= 2 x - 9 0^{\circ}

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

Join

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

2 x - 9 0^{\circ}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

2 \cdot 2 = 4

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

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

Apply the fraction rule:

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

= \frac{4 x - 18 0 ^{\circ}}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 x - 18 0 ^{\circ}}{4}

= 2 sin (4 5^{\circ}) cos (\frac{4 x - 18 0 ^{\circ}}{4})

Simplify

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

sin (4 5^{\circ})

Use the following trivial identity:

sin (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}

= \frac{2}{2}

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

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

2 cos (\frac{4 x - 18 0 ^{\circ}}{4}) = 1

Divide both sides by

2

2 cos (\frac{4 x - 18 0 ^{\circ}}{4}) = 1

Divide both sides by

2

\frac{2 cos ( \frac{4 x - 18 0 ^{\circ}}{4} )}{2} = \frac{1}{2}

Simplify

\frac{2 cos ( \frac{4 x - 18 0 ^{\circ}}{4} )}{2} = \frac{1}{2}

Simplify

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

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

Cancel the common factor:

2

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

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}

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

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

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

General solutions for

cos (\frac{4 x - 18 0 ^{\circ}}{4}) = \frac{2}{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{4 x - 18 0 ^{\circ}}{4} = 4 5^{\circ} + 36 0^{\circ} n, \frac{4 x - 18 0 ^{\circ}}{4} = 31 5^{\circ} + 36 0^{\circ} n

\frac{4 x - 18 0 ^{\circ}}{4} = 4 5^{\circ} + 36 0^{\circ} n, \frac{4 x - 18 0 ^{\circ}}{4} = 31 5^{\circ} + 36 0^{\circ} n

Solve

\frac{4 x - 18 0 ^{\circ}}{4} = 4 5^{\circ} + 36 0^{\circ} n : x = 9 0^{\circ} + 36 0^{\circ} n

\frac{4 x - 18 0 ^{\circ}}{4} = 4 5^{\circ} + 36 0^{\circ} n

Multiply both sides by

4

\frac{4 x - 18 0 ^{\circ}}{4} = 4 5^{\circ} + 36 0^{\circ} n

Multiply both sides by

4

\frac{4 ( 4 x - 18 0 ^{\circ} )}{4} = 4 \cdot 4 5^{\circ} + 4 \cdot 36 0^{\circ} n

Simplify

\frac{4 ( 4 x - 18 0 ^{\circ} )}{4} = 4 \cdot 4 5^{\circ} + 4 \cdot 36 0^{\circ} n

Simplify

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

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

Divide the numbers:

\frac{4}{4} = 1

= 4 x - 18 0^{\circ}

Simplify

4 \cdot 4 5^{\circ} + 4 \cdot 36 0^{\circ} n : 18 0^{\circ} + 144 0^{\circ} n

4 \cdot 4 5^{\circ} + 4 \cdot 36 0^{\circ} n

4 \cdot 4 5^{\circ} = 18 0^{\circ}

4 \cdot 4 5^{\circ}

Multiply fractions:

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

= 18 0^{\circ}

Cancel the common factor:

4

= 18 0^{\circ}

4 \cdot 36 0^{\circ} n = 144 0^{\circ} n

4 \cdot 36 0^{\circ} n

Multiply the numbers:

4 \cdot 2 = 8

= 144 0^{\circ} n

= 18 0^{\circ} + 144 0^{\circ} n

4 x - 18 0^{\circ} = 18 0^{\circ} + 144 0^{\circ} n

4 x - 18 0^{\circ} = 18 0^{\circ} + 144 0^{\circ} n

4 x - 18 0^{\circ} = 18 0^{\circ} + 144 0^{\circ} n

Move

18 0^{\circ}

to the right side

4 x - 18 0^{\circ} = 18 0^{\circ} + 144 0^{\circ} n

Add

18 0^{\circ}

to both sides

4 x - 18 0^{\circ} + 18 0^{\circ} = 18 0^{\circ} + 144 0^{\circ} n + 18 0^{\circ}

Simplify

4 x = 36 0^{\circ} + 144 0^{\circ} n

4 x = 36 0^{\circ} + 144 0^{\circ} n

Divide both sides by

4

4 x = 36 0^{\circ} + 144 0^{\circ} n

Divide both sides by

4

\frac{4 x}{4} = 9 0^{\circ} + \frac{144 0 ^{\circ} n}{4}

Simplify

\frac{4 x}{4} = 9 0^{\circ} + \frac{144 0 ^{\circ} n}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

9 0^{\circ} + \frac{144 0 ^{\circ} n}{4} : 9 0^{\circ} + 36 0^{\circ} n

9 0^{\circ} + \frac{144 0 ^{\circ} n}{4}

Cancel

9 0^{\circ} : 9 0^{\circ}

9 0^{\circ}

Cancel the common factor:

2

= 9 0^{\circ}

= 9 0^{\circ} + \frac{144 0 ^{\circ} n}{4}

Divide the numbers:

\frac{8}{4} = 2

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

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

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

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

Solve

\frac{4 x - 18 0 ^{\circ}}{4} = 31 5^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} + 36 0^{\circ} n

\frac{4 x - 18 0 ^{\circ}}{4} = 31 5^{\circ} + 36 0^{\circ} n

Multiply both sides by

4

\frac{4 x - 18 0 ^{\circ}}{4} = 31 5^{\circ} + 36 0^{\circ} n

Multiply both sides by

4

\frac{4 ( 4 x - 18 0 ^{\circ} )}{4} = 4 \cdot 31 5^{\circ} + 4 \cdot 36 0^{\circ} n

Simplify

\frac{4 ( 4 x - 18 0 ^{\circ} )}{4} = 4 \cdot 31 5^{\circ} + 4 \cdot 36 0^{\circ} n

Simplify

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

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

Divide the numbers:

\frac{4}{4} = 1

= 4 x - 18 0^{\circ}

Simplify

4 \cdot 31 5^{\circ} + 4 \cdot 36 0^{\circ} n : 126 0^{\circ} + 144 0^{\circ} n

4 \cdot 31 5^{\circ} + 4 \cdot 36 0^{\circ} n

4 \cdot 31 5^{\circ} = 126 0^{\circ}

4 \cdot 31 5^{\circ}

Multiply fractions:

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

= 126 0^{\circ}

Cancel the common factor:

4

= 126 0^{\circ}

4 \cdot 36 0^{\circ} n = 144 0^{\circ} n

4 \cdot 36 0^{\circ} n

Multiply the numbers:

4 \cdot 2 = 8

= 144 0^{\circ} n

= 126 0^{\circ} + 144 0^{\circ} n

4 x - 18 0^{\circ} = 126 0^{\circ} + 144 0^{\circ} n

4 x - 18 0^{\circ} = 126 0^{\circ} + 144 0^{\circ} n

4 x - 18 0^{\circ} = 126 0^{\circ} + 144 0^{\circ} n

Move

18 0^{\circ}

to the right side

4 x - 18 0^{\circ} = 126 0^{\circ} + 144 0^{\circ} n

Add

18 0^{\circ}

to both sides

4 x - 18 0^{\circ} + 18 0^{\circ} = 126 0^{\circ} + 144 0^{\circ} n + 18 0^{\circ}

Simplify

4 x = 144 0^{\circ} + 144 0^{\circ} n

4 x = 144 0^{\circ} + 144 0^{\circ} n

Divide both sides by

4

4 x = 144 0^{\circ} + 144 0^{\circ} n

Divide both sides by

4

\frac{4 x}{4} = 36 0^{\circ} + \frac{144 0 ^{\circ} n}{4}

Simplify

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

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

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

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)+sin(90-x)=sin(90) ?
The general solution for sin(x)+sin(90-x)=sin(90) is x=90+360n,x=360+360n