What is the general solution for cos(2x)=sin(70+x) ?

The general solution for cos(2x)=sin(70+x) is x=(3240n+180}{27},x=-\frac{180+3240n)/9

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cos(2x)=sin(70+x)

Solution

cos (2 x) = sin (7 0^{\circ} + x)

Solution

x = \frac{324 0 ^{\circ} n + 18 0 ^{\circ}}{27}, x = - \frac{18 0 ^{\circ} + 324 0 ^{\circ} n}{9}

Radians

x = \frac{π}{27} + \frac{18 π}{27} n, x = - \frac{π}{9} - \frac{18 π}{9} n

Solution steps

cos (2 x) = sin (7 0^{\circ} + x)

Rewrite using trig identities

cos (2 x) = sin (7 0^{\circ} + x)

Use the following identity:

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

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

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

Apply trig inverse properties

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

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

7 0^{\circ} + x = 9 0^{\circ} - 2 x + 36 0^{\circ} n, 7 0^{\circ} + x = 18 0^{\circ} - (9 0^{\circ} - 2 x) + 36 0^{\circ} n

7 0^{\circ} + x = 9 0^{\circ} - 2 x + 36 0^{\circ} n, 7 0^{\circ} + x = 18 0^{\circ} - (9 0^{\circ} - 2 x) + 36 0^{\circ} n

7 0^{\circ} + x = 9 0^{\circ} - 2 x + 36 0^{\circ} n : x = \frac{324 0 ^{\circ} n + 18 0 ^{\circ}}{27}

7 0^{\circ} + x = 9 0^{\circ} - 2 x + 36 0^{\circ} n

Move

7 0^{\circ}

to the right side

7 0^{\circ} + x = 9 0^{\circ} - 2 x + 36 0^{\circ} n

Subtract

7 0^{\circ}

from both sides

7 0^{\circ} + x - 7 0^{\circ} = 9 0^{\circ} - 2 x + 36 0^{\circ} n - 7 0^{\circ}

Simplify

7 0^{\circ} + x - 7 0^{\circ} = 9 0^{\circ} - 2 x + 36 0^{\circ} n - 7 0^{\circ}

Simplify

7 0^{\circ} + x - 7 0^{\circ} : x

7 0^{\circ} + x - 7 0^{\circ}

Add similar elements:

7 0^{\circ} - 7 0^{\circ} = 0

= x

Simplify

9 0^{\circ} - 2 x + 36 0^{\circ} n - 7 0^{\circ} : - 2 x + 36 0^{\circ} n + 2 0^{\circ}

9 0^{\circ} - 2 x + 36 0^{\circ} n - 7 0^{\circ}

Group like terms

= - 2 x + 36 0^{\circ} n + 9 0^{\circ} - 7 0^{\circ}

Least Common Multiplier of

2, 18 : 18

2, 18

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

18 : 2 \cdot 3 \cdot 3

18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3 \cdot 3

Multiply each factor the greatest number of times it occurs in either

2

18

= 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

9 0^{\circ} :

multiply the denominator and numerator by

9

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

= 9 0^{\circ} - 7 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{18 0 ^{\circ} 9 - 126 0 ^{\circ}}{18}

Add similar elements:

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

= 2 0^{\circ}

Cancel the common factor:

2

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

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

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

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

Move

2 x

to the left side

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

Add

2 x

to both sides

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

Simplify

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

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

Divide both sides by

3

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

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{36 0 ^{\circ} n}{3} + \frac{2 0 ^{\circ}}{3} : \frac{324 0 ^{\circ} n + 18 0 ^{\circ}}{27}

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

Apply rule

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

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

Join

36 0^{\circ} n + 2 0^{\circ} : \frac{324 0 ^{\circ} n + 18 0 ^{\circ}}{9}

36 0^{\circ} n + 2 0^{\circ}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

2 \cdot 9 = 18

= \frac{324 0 ^{\circ} n + 18 0 ^{\circ}}{9}

= \frac{\frac{324 0 ^{\circ} n + 18 0 ^{\circ}}{9}}{3}

Apply the fraction rule:

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

= \frac{324 0 ^{\circ} n + 18 0 ^{\circ}}{9 \cdot 3}

Multiply the numbers:

9 \cdot 3 = 27

= \frac{324 0 ^{\circ} n + 18 0 ^{\circ}}{27}

x = \frac{324 0 ^{\circ} n + 18 0 ^{\circ}}{27}

x = \frac{324 0 ^{\circ} n + 18 0 ^{\circ}}{27}

x = \frac{324 0 ^{\circ} n + 18 0 ^{\circ}}{27}

7 0^{\circ} + x = 18 0^{\circ} - (9 0^{\circ} - 2 x) + 36 0^{\circ} n : x = - \frac{18 0 ^{\circ} + 324 0 ^{\circ} n}{9}

7 0^{\circ} + x = 18 0^{\circ} - (9 0^{\circ} - 2 x) + 36 0^{\circ} n

Expand

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

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

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

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

Distribute parentheses

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

Apply minus-plus rules

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

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

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

7 0^{\circ} + x = 18 0^{\circ} - 9 0^{\circ} + 2 x + 36 0^{\circ} n

Move

7 0^{\circ}

to the right side

7 0^{\circ} + x = 18 0^{\circ} - 9 0^{\circ} + 2 x + 36 0^{\circ} n

Subtract

7 0^{\circ}

from both sides

7 0^{\circ} + x - 7 0^{\circ} = 18 0^{\circ} - 9 0^{\circ} + 2 x + 36 0^{\circ} n - 7 0^{\circ}

Simplify

7 0^{\circ} + x - 7 0^{\circ} = 18 0^{\circ} - 9 0^{\circ} + 2 x + 36 0^{\circ} n - 7 0^{\circ}

Simplify

7 0^{\circ} + x - 7 0^{\circ} : x

7 0^{\circ} + x - 7 0^{\circ}

Add similar elements:

7 0^{\circ} - 7 0^{\circ} = 0

= x

Simplify

18 0^{\circ} - 9 0^{\circ} + 2 x + 36 0^{\circ} n - 7 0^{\circ} : 2 x + 18 0^{\circ} + 36 0^{\circ} n - 16 0^{\circ}

18 0^{\circ} - 9 0^{\circ} + 2 x + 36 0^{\circ} n - 7 0^{\circ}

Group like terms

= 2 x + 18 0^{\circ} + 36 0^{\circ} n - 9 0^{\circ} - 7 0^{\circ}

Least Common Multiplier of

2, 18 : 18

2, 18

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

18 : 2 \cdot 3 \cdot 3

18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3 \cdot 3

Multiply each factor the greatest number of times it occurs in either

2

18

= 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

9 0^{\circ} :

multiply the denominator and numerator by

9

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

= - 9 0^{\circ} - 7 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{- 18 0 ^{\circ} 9 - 126 0 ^{\circ}}{18}

Add similar elements:

- 162 0^{\circ} - 126 0^{\circ} = - 288 0^{\circ}

= \frac{- 288 0 ^{\circ}}{18}

Apply the fraction rule:

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

= - 16 0^{\circ}

Cancel the common factor:

2

= 2 x + 18 0^{\circ} + 36 0^{\circ} n - 16 0^{\circ}

x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 16 0^{\circ}

x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 16 0^{\circ}

x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 16 0^{\circ}

Move

2 x

to the left side

x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 16 0^{\circ}

Subtract

2 x

from both sides

x - 2 x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 16 0^{\circ} - 2 x

Simplify

- x = 18 0^{\circ} + 36 0^{\circ} n - 16 0^{\circ}

- x = 18 0^{\circ} + 36 0^{\circ} n - 16 0^{\circ}

Divide both sides by

- 1

- x = 18 0^{\circ} + 36 0^{\circ} n - 16 0^{\circ}

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{18 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{16 0 ^{\circ}}{- 1}

Simplify

\frac{- x}{- 1} = \frac{18 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{16 0 ^{\circ}}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{18 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{16 0 ^{\circ}}{- 1} : - \frac{18 0 ^{\circ} + 324 0 ^{\circ} n}{9}

\frac{18 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{16 0 ^{\circ}}{- 1}

Apply rule

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

= \frac{18 0 ^{\circ} + 36 0 ^{\circ} n - 16 0 ^{\circ}}{- 1}

Apply the fraction rule:

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

= - \frac{18 0 ^{\circ} + 36 0 ^{\circ} n - 16 0 ^{\circ}}{1}

Join

18 0^{\circ} + 36 0^{\circ} n - 16 0^{\circ} : \frac{18 0 ^{\circ} + 324 0 ^{\circ} n}{9}

18 0^{\circ} + 36 0^{\circ} n - 16 0^{\circ}

Convert element to fraction:

18 0^{\circ} = 18 0^{\circ}, 36 0^{\circ} n = \frac{36 0 ^{\circ} n 9}{9}

= 18 0^{\circ} + \frac{36 0 ^{\circ} n \cdot 9}{9} - 16 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{18 0 ^{\circ} 9 + 36 0 ^{\circ} n \cdot 9 - 144 0 ^{\circ}}{9}

18 0^{\circ} 9 + 36 0^{\circ} n \cdot 9 - 144 0^{\circ} = 18 0^{\circ} + 324 0^{\circ} n

18 0^{\circ} 9 + 36 0^{\circ} n \cdot 9 - 144 0^{\circ}

Add similar elements:

162 0^{\circ} - 144 0^{\circ} = 18 0^{\circ}

= 18 0^{\circ} + 2 \cdot 162 0^{\circ} n

Multiply the numbers:

2 \cdot 9 = 18

= 18 0^{\circ} + 324 0^{\circ} n

= \frac{18 0 ^{\circ} + 324 0 ^{\circ} n}{9}

= - \frac{\frac{18 0 ^{\circ} + 324 0 ^{\circ} n}{9}}{1}

Apply the fraction rule:

\frac{a}{1} = a

= - \frac{18 0 ^{\circ} + 324 0 ^{\circ} n}{9}

x = - \frac{18 0 ^{\circ} + 324 0 ^{\circ} n}{9}

x = - \frac{18 0 ^{\circ} + 324 0 ^{\circ} n}{9}

x = - \frac{18 0 ^{\circ} + 324 0 ^{\circ} n}{9}

x = \frac{324 0 ^{\circ} n + 18 0 ^{\circ}}{27}, x = - \frac{18 0 ^{\circ} + 324 0 ^{\circ} n}{9}

x = \frac{324 0 ^{\circ} n + 18 0 ^{\circ}}{27}, x = - \frac{18 0 ^{\circ} + 324 0 ^{\circ} n}{9}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2x)=sin(70+x) ?
The general solution for cos(2x)=sin(70+x) is x=(3240n+180}{27},x=-\frac{180+3240n)/9