What is the general solution for cos(x/2+20)=sin(x) ?

The general solution for cos(x/2+20)=sin(x) is x=(1260+6480n}{27},x=\frac{1980+6480n)/9

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Popular Trigonometry >

cos(x/2+20)=sin(x)

Solution

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

Solution

x = \frac{126 0 ^{\circ} + 648 0 ^{\circ} n}{27}, x = \frac{198 0 ^{\circ} + 648 0 ^{\circ} n}{9}

Radians

x = \frac{7 π}{27} + \frac{36 π}{27} n, x = \frac{11 π}{9} + \frac{36 π}{9} n

Solution steps

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

Rewrite using trig identities

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

Use the following identity:

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

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

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

Apply trig inverse properties

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

sin (x) = sin (y) \Rightarrow x = y + 2 π n, x = π - y + 2 π n

x = 9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ}) + 36 0^{\circ} n, x = 18 0^{\circ} - (9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ})) + 36 0^{\circ} n

x = 9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ}) + 36 0^{\circ} n, x = 18 0^{\circ} - (9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ})) + 36 0^{\circ} n

x = 9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ}) + 36 0^{\circ} n : x = \frac{126 0 ^{\circ} + 648 0 ^{\circ} n}{27}

x = 9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ}) + 36 0^{\circ} n

Move

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

to the left side

x = 9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ}) + 36 0^{\circ} n

Add

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

to both sides

x + \frac{x}{2} + 2 0^{\circ} = 9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ}) + 36 0^{\circ} n + \frac{x}{2} + 2 0^{\circ}

Simplify

x + \frac{x}{2} + 2 0^{\circ} = 9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ}) + 36 0^{\circ} n + \frac{x}{2} + 2 0^{\circ}

Simplify

x + \frac{x}{2} + 2 0^{\circ} : \frac{27 x + 36 0 ^{\circ}}{18}

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

Convert element to fraction:

x = \frac{x}{1}

= \frac{x}{2} + 2 0^{\circ} + \frac{x}{1}

Least Common Multiplier of

2, 9, 1 : 18

2, 9, 1

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

Prime factorization of

1

Compute a number comprised of factors that appear in at least one of the following:

2, 9, 1

= 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 3 \cdot 3 = 18

= 18

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

18

For

\frac{x}{2} :

multiply the denominator and numerator by

9

\frac{x}{2} = \frac{x \cdot 9}{2 \cdot 9} = \frac{x \cdot 9}{18}

For

2 0^{\circ} :

multiply the denominator and numerator by

2

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

For

\frac{x}{1} :

multiply the denominator and numerator by

18

\frac{x}{1} = \frac{x \cdot 18}{1 \cdot 18} = \frac{x \cdot 18}{18}

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

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} 2 + x \cdot 18}{18}

x \cdot 9 + 18 0^{\circ} 2 + x \cdot 18 = 27 x + 36 0^{\circ}

x \cdot 9 + 18 0^{\circ} 2 + x \cdot 18

Group like terms

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

Add similar elements:

9 x + 18 x = 27 x

= 27 x + 36 0^{\circ}

= \frac{27 x + 36 0 ^{\circ}}{18}

Simplify

9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ}) + 36 0^{\circ} n + \frac{x}{2} + 2 0^{\circ} : 9 0^{\circ} + 36 0^{\circ} n

9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ}) + 36 0^{\circ} n + \frac{x}{2} + 2 0^{\circ}

Add similar elements:

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

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

\frac{27 x + 36 0 ^{\circ}}{18} = 9 0^{\circ} + 36 0^{\circ} n

\frac{27 x + 36 0 ^{\circ}}{18} = 9 0^{\circ} + 36 0^{\circ} n

\frac{27 x + 36 0 ^{\circ}}{18} = 9 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

18

\frac{27 x + 36 0 ^{\circ}}{18} = 9 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

18

\frac{18 ( 27 x + 36 0 ^{\circ} )}{18} = 18 \cdot 9 0^{\circ} + 18 \cdot 36 0^{\circ} n

Simplify

\frac{18 ( 27 x + 36 0 ^{\circ} )}{18} = 18 \cdot 9 0^{\circ} + 18 \cdot 36 0^{\circ} n

Simplify

\frac{18 ( 27 x + 36 0 ^{\circ} )}{18} : 27 x + 36 0^{\circ}

\frac{18 ( 27 x + 36 0 ^{\circ} )}{18}

Divide the numbers:

\frac{18}{18} = 1

= 27 x + 36 0^{\circ}

Simplify

18 \cdot 9 0^{\circ} + 18 \cdot 36 0^{\circ} n : 162 0^{\circ} + 648 0^{\circ} n

18 \cdot 9 0^{\circ} + 18 \cdot 36 0^{\circ} n

18 \cdot 9 0^{\circ} = 162 0^{\circ}

18 \cdot 9 0^{\circ}

Multiply fractions:

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

= 162 0^{\circ}

Divide the numbers:

\frac{18}{2} = 9

= 162 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

= 162 0^{\circ} + 648 0^{\circ} n

27 x + 36 0^{\circ} = 162 0^{\circ} + 648 0^{\circ} n

27 x + 36 0^{\circ} = 162 0^{\circ} + 648 0^{\circ} n

27 x + 36 0^{\circ} = 162 0^{\circ} + 648 0^{\circ} n

Move

36 0^{\circ}

to the right side

27 x + 36 0^{\circ} = 162 0^{\circ} + 648 0^{\circ} n

Subtract

36 0^{\circ}

from both sides

27 x + 36 0^{\circ} - 36 0^{\circ} = 162 0^{\circ} + 648 0^{\circ} n - 36 0^{\circ}

Simplify

27 x = 126 0^{\circ} + 648 0^{\circ} n

27 x = 126 0^{\circ} + 648 0^{\circ} n

Divide both sides by

27

27 x = 126 0^{\circ} + 648 0^{\circ} n

Divide both sides by

27

\frac{27 x}{27} = 46.66666 \dots^{\circ} + \frac{648 0 ^{\circ} n}{27}

Simplify

\frac{27 x}{27} = 46.66666 \dots^{\circ} + \frac{648 0 ^{\circ} n}{27}

Simplify

\frac{27 x}{27} : x

\frac{27 x}{27}

Divide the numbers:

\frac{27}{27} = 1

= x

Simplify

46.66666 \dots^{\circ} + \frac{648 0 ^{\circ} n}{27} : \frac{126 0 ^{\circ} + 648 0 ^{\circ} n}{27}

46.66666 \dots^{\circ} + \frac{648 0 ^{\circ} n}{27}

Apply rule

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

= \frac{126 0 ^{\circ} + 648 0 ^{\circ} n}{27}

x = \frac{126 0 ^{\circ} + 648 0 ^{\circ} n}{27}

x = \frac{126 0 ^{\circ} + 648 0 ^{\circ} n}{27}

x = \frac{126 0 ^{\circ} + 648 0 ^{\circ} n}{27}

x = 18 0^{\circ} - (9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ})) + 36 0^{\circ} n : x = \frac{198 0 ^{\circ} + 648 0 ^{\circ} n}{9}

x = 18 0^{\circ} - (9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ})) + 36 0^{\circ} n

Move

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

to the left side

x = 18 0^{\circ} - (9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ})) + 36 0^{\circ} n

Add

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

to both sides

x + 9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ}) = 18 0^{\circ} - (9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ})) + 36 0^{\circ} n + 9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ})

Simplify

x + 9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ}) = 18 0^{\circ} - (9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ})) + 36 0^{\circ} n + 9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ})

Simplify

x + 9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ}) : \frac{9 x + 126 0 ^{\circ}}{18}

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

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

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

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

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

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

Simplify

x + 9 0^{\circ} - \frac{x}{2} - 2 0^{\circ} : \frac{9 x + 126 0 ^{\circ}}{18}

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

Group like terms

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

Combine the fractions

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

Apply rule

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

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

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

Convert element to fraction:

x = \frac{x}{1}

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

Least Common Multiplier of

2, 9, 1 : 18

2, 9, 1

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

Prime factorization of

1

Compute a number comprised of factors that appear in at least one of the following:

2, 9, 1

= 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 3 \cdot 3 = 18

= 18

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

18

For

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

multiply the denominator and numerator by

9

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

For

2 0^{\circ} :

multiply the denominator and numerator by

2

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

For

\frac{x}{1} :

multiply the denominator and numerator by

18

\frac{x}{1} = \frac{x \cdot 18}{1 \cdot 18} = \frac{x \cdot 18}{18}

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

Since the denominators are equal, combine the fractions:

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

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

Expand

(- x + 18 0^{\circ}) \cdot 9 - 18 0^{\circ} 2 + x \cdot 18 : 9 x + 126 0^{\circ}

(- x + 18 0^{\circ}) \cdot 9 - 18 0^{\circ} 2 + x \cdot 18

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

Expand

9 (- x + 18 0^{\circ}) : - 9 x + 162 0^{\circ}

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

Apply the distributive law:

a (b + c) = ab + a c

a = 9, b = - x, c = 18 0^{\circ}

= 9 (- x) + 162 0^{\circ}

Apply minus-plus rules

+ (- a) = - a

= - 9 x + 162 0^{\circ}

= - 9 x + 162 0^{\circ} - 18 0^{\circ} 2 + x \cdot 18

Simplify

- 9 x + 162 0^{\circ} - 18 0^{\circ} 2 + x \cdot 18 : 9 x + 126 0^{\circ}

- 9 x + 162 0^{\circ} - 18 0^{\circ} 2 + x \cdot 18

Group like terms

= - 9 x + 18 x + 162 0^{\circ} - 36 0^{\circ}

Add similar elements:

- 9 x + 18 x = 9 x

= 9 x + 162 0^{\circ} - 36 0^{\circ}

Add similar elements:

162 0^{\circ} - 36 0^{\circ} = 126 0^{\circ}

= 9 x + 126 0^{\circ}

= 9 x + 126 0^{\circ}

= \frac{9 x + 126 0 ^{\circ}}{18}

= \frac{9 x + 126 0 ^{\circ}}{18}

Simplify

18 0^{\circ} - (9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ})) + 36 0^{\circ} n + 9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ}) : 18 0^{\circ} + 36 0^{\circ} n

18 0^{\circ} - (9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ})) + 36 0^{\circ} n + 9 0^{\circ} - (\frac{x}{2} + 2 0^{\circ})

Add similar elements:

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

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

\frac{9 x + 126 0 ^{\circ}}{18} = 18 0^{\circ} + 36 0^{\circ} n

\frac{9 x + 126 0 ^{\circ}}{18} = 18 0^{\circ} + 36 0^{\circ} n

\frac{9 x + 126 0 ^{\circ}}{18} = 18 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

18

\frac{9 x + 126 0 ^{\circ}}{18} = 18 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

18

\frac{18 ( 9 x + 126 0 ^{\circ} )}{18} = 324 0^{\circ} + 18 \cdot 36 0^{\circ} n

Simplify

9 x + 126 0^{\circ} = 324 0^{\circ} + 648 0^{\circ} n

9 x + 126 0^{\circ} = 324 0^{\circ} + 648 0^{\circ} n

Move

126 0^{\circ}

to the right side

9 x + 126 0^{\circ} = 324 0^{\circ} + 648 0^{\circ} n

Subtract

126 0^{\circ}

from both sides

9 x + 126 0^{\circ} - 126 0^{\circ} = 324 0^{\circ} + 648 0^{\circ} n - 126 0^{\circ}

Simplify

9 x = 198 0^{\circ} + 648 0^{\circ} n

9 x = 198 0^{\circ} + 648 0^{\circ} n

Divide both sides by

9

9 x = 198 0^{\circ} + 648 0^{\circ} n

Divide both sides by

9

\frac{9 x}{9} = 22 0^{\circ} + \frac{648 0 ^{\circ} n}{9}

Simplify

\frac{9 x}{9} = 22 0^{\circ} + \frac{648 0 ^{\circ} n}{9}

Simplify

\frac{9 x}{9} : x

\frac{9 x}{9}

Divide the numbers:

\frac{9}{9} = 1

= x

Simplify

22 0^{\circ} + \frac{648 0 ^{\circ} n}{9} : \frac{198 0 ^{\circ} + 648 0 ^{\circ} n}{9}

22 0^{\circ} + \frac{648 0 ^{\circ} n}{9}

Apply rule

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

= \frac{198 0 ^{\circ} + 648 0 ^{\circ} n}{9}

x = \frac{198 0 ^{\circ} + 648 0 ^{\circ} n}{9}

x = \frac{198 0 ^{\circ} + 648 0 ^{\circ} n}{9}

x = \frac{198 0 ^{\circ} + 648 0 ^{\circ} n}{9}

x = \frac{126 0 ^{\circ} + 648 0 ^{\circ} n}{27}, x = \frac{198 0 ^{\circ} + 648 0 ^{\circ} n}{9}

x = \frac{126 0 ^{\circ} + 648 0 ^{\circ} n}{27}, x = \frac{198 0 ^{\circ} + 648 0 ^{\circ} n}{9}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos(x/2+20)=sin(x) ?
The general solution for cos(x/2+20)=sin(x) is x=(1260+6480n}{27},x=\frac{1980+6480n)/9