What is the general solution for cos(50+x)=sin(2x-6) ?

The general solution for cos(50+x)=sin(2x-6) is x=(3240n+360+54)/(27),x=180-40+360n+6

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cos(50+x)=sin(2x-6)

Solution

cos (5 0^{\circ} + x) = sin (2 x - 6)

Solution

x = \frac{324 0 ^{\circ} n + 36 0 ^{\circ} + 54}{27}, x = 18 0^{\circ} - 4 0^{\circ} + 36 0^{\circ} n + 6

Radians

x = \frac{2 π}{27} + 2 + \frac{18 π}{27} n, x = π - \frac{2 π}{9} + 6 + 2 πn

Solution steps

cos (5 0^{\circ} + x) = sin (2 x - 6)

Rewrite using trig identities

cos (5 0^{\circ} + x) = sin (2 x - 6)

Use the following identity:

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

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

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

Apply trig inverse properties

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

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

2 x - 6 = 9 0^{\circ} - (5 0^{\circ} + x) + 36 0^{\circ} n, 2 x - 6 = 18 0^{\circ} - (9 0^{\circ} - (5 0^{\circ} + x)) + 36 0^{\circ} n

2 x - 6 = 9 0^{\circ} - (5 0^{\circ} + x) + 36 0^{\circ} n, 2 x - 6 = 18 0^{\circ} - (9 0^{\circ} - (5 0^{\circ} + x)) + 36 0^{\circ} n

2 x - 6 = 9 0^{\circ} - (5 0^{\circ} + x) + 36 0^{\circ} n : x = \frac{324 0 ^{\circ} n + 36 0 ^{\circ} + 54}{27}

2 x - 6 = 9 0^{\circ} - (5 0^{\circ} + x) + 36 0^{\circ} n

Expand

9 0^{\circ} - (5 0^{\circ} + x) + 36 0^{\circ} n : - x + 36 0^{\circ} n + 4 0^{\circ}

9 0^{\circ} - (5 0^{\circ} + x) + 36 0^{\circ} n

- (5 0^{\circ} + x) : - 5 0^{\circ} - x

- (5 0^{\circ} + x)

Distribute parentheses

= - (5 0^{\circ}) - (x)

Apply minus-plus rules

+ (- a) = - a

= - 5 0^{\circ} - x

= 9 0^{\circ} - 5 0^{\circ} - x + 36 0^{\circ} n

Simplify

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

9 0^{\circ} - 5 0^{\circ} - x + 36 0^{\circ} n

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} - 5 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 - 90 0 ^{\circ}}{18}

Add similar elements:

162 0^{\circ} - 90 0^{\circ} = 72 0^{\circ}

= 4 0^{\circ}

Cancel the common factor:

2

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

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

2 x - 6 = - x + 36 0^{\circ} n + 4 0^{\circ}

Move

6

to the right side

2 x - 6 = - x + 36 0^{\circ} n + 4 0^{\circ}

Add

6

to both sides

2 x - 6 + 6 = - x + 36 0^{\circ} n + 4 0^{\circ} + 6

Simplify

2 x = - x + 36 0^{\circ} n + 4 0^{\circ} + 6

2 x = - x + 36 0^{\circ} n + 4 0^{\circ} + 6

Move

x

to the left side

2 x = - x + 36 0^{\circ} n + 4 0^{\circ} + 6

Add

x

to both sides

2 x + x = - x + 36 0^{\circ} n + 4 0^{\circ} + 6 + x

Simplify

3 x = 36 0^{\circ} n + 4 0^{\circ} + 6

3 x = 36 0^{\circ} n + 4 0^{\circ} + 6

Divide both sides by

3

3 x = 36 0^{\circ} n + 4 0^{\circ} + 6

Divide both sides by

3

\frac{3 x}{3} = \frac{36 0 ^{\circ} n}{3} + \frac{4 0 ^{\circ}}{3} + \frac{6}{3}

Simplify

\frac{3 x}{3} = \frac{36 0 ^{\circ} n}{3} + \frac{4 0 ^{\circ}}{3} + \frac{6}{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{4 0 ^{\circ}}{3} + \frac{6}{3} : \frac{324 0 ^{\circ} n + 36 0 ^{\circ} + 54}{27}

\frac{36 0 ^{\circ} n}{3} + \frac{4 0 ^{\circ}}{3} + \frac{6}{3}

Apply rule

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

= \frac{36 0 ^{\circ} n + 4 0 ^{\circ} + 6}{3}

Join

36 0^{\circ} n + 4 0^{\circ} + 6 : \frac{324 0 ^{\circ} n + 36 0 ^{\circ} + 54}{9}

36 0^{\circ} n + 4 0^{\circ} + 6

Convert element to fraction:

36 0^{\circ} n = \frac{36 0 ^{\circ} n 9}{9}, 6 = \frac{6 \cdot 9}{9}

= \frac{36 0 ^{\circ} n \cdot 9}{9} + 4 0^{\circ} + \frac{6 \cdot 9}{9}

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 + 36 0 ^{\circ} + 6 \cdot 9}{9}

36 0^{\circ} n \cdot 9 + 36 0^{\circ} + 6 \cdot 9 = 324 0^{\circ} n + 36 0^{\circ} + 54

36 0^{\circ} n \cdot 9 + 36 0^{\circ} + 6 \cdot 9

Multiply the numbers:

2 \cdot 9 = 18

= 324 0^{\circ} n + 36 0^{\circ} + 6 \cdot 9

Multiply the numbers:

6 \cdot 9 = 54

= 324 0^{\circ} n + 36 0^{\circ} + 54

= \frac{324 0 ^{\circ} n + 36 0 ^{\circ} + 54}{9}

= \frac{\frac{324 0 ^{\circ} n + 36 0 ^{\circ} + 54}{9}}{3}

Apply the fraction rule:

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

= \frac{324 0 ^{\circ} n + 36 0 ^{\circ} + 54}{9 \cdot 3}

Multiply the numbers:

9 \cdot 3 = 27

= \frac{324 0 ^{\circ} n + 36 0 ^{\circ} + 54}{27}

x = \frac{324 0 ^{\circ} n + 36 0 ^{\circ} + 54}{27}

x = \frac{324 0 ^{\circ} n + 36 0 ^{\circ} + 54}{27}

x = \frac{324 0 ^{\circ} n + 36 0 ^{\circ} + 54}{27}

2 x - 6 = 18 0^{\circ} - (9 0^{\circ} - (5 0^{\circ} + x)) + 36 0^{\circ} n : x = 18 0^{\circ} - 4 0^{\circ} + 36 0^{\circ} n + 6

2 x - 6 = 18 0^{\circ} - (9 0^{\circ} - (5 0^{\circ} + x)) + 36 0^{\circ} n

Expand

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

18 0^{\circ} - (9 0^{\circ} - (5 0^{\circ} + x)) + 36 0^{\circ} n

Expand

9 0^{\circ} - (5 0^{\circ} + x) : - x + 4 0^{\circ}

9 0^{\circ} - (5 0^{\circ} + x)

- (5 0^{\circ} + x) : - 5 0^{\circ} - x

- (5 0^{\circ} + x)

Distribute parentheses

= - (5 0^{\circ}) - (x)

Apply minus-plus rules

+ (- a) = - a

= - 5 0^{\circ} - x

= 9 0^{\circ} - 5 0^{\circ} - x

Simplify

9 0^{\circ} - 5 0^{\circ} - x : - x + 4 0^{\circ}

9 0^{\circ} - 5 0^{\circ} - x

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} - 5 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 - 90 0 ^{\circ}}{18}

Add similar elements:

162 0^{\circ} - 90 0^{\circ} = 72 0^{\circ}

= 4 0^{\circ}

Cancel the common factor:

2

= - x + 4 0^{\circ}

= - x + 4 0^{\circ}

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

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

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

Distribute parentheses

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

Apply minus-plus rules

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

= x - 4 0^{\circ}

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

2 x - 6 = 18 0^{\circ} + x - 4 0^{\circ} + 36 0^{\circ} n

Move

6

to the right side

2 x - 6 = 18 0^{\circ} + x - 4 0^{\circ} + 36 0^{\circ} n

Add

6

to both sides

2 x - 6 + 6 = 18 0^{\circ} + x - 4 0^{\circ} + 36 0^{\circ} n + 6

Simplify

2 x = 18 0^{\circ} + x - 4 0^{\circ} + 36 0^{\circ} n + 6

2 x = 18 0^{\circ} + x - 4 0^{\circ} + 36 0^{\circ} n + 6

Move

x

to the left side

2 x = 18 0^{\circ} + x - 4 0^{\circ} + 36 0^{\circ} n + 6

Subtract

x

from both sides

2 x - x = 18 0^{\circ} + x - 4 0^{\circ} + 36 0^{\circ} n + 6 - x

Simplify

x = 18 0^{\circ} - 4 0^{\circ} + 36 0^{\circ} n + 6

x = 18 0^{\circ} - 4 0^{\circ} + 36 0^{\circ} n + 6

x = \frac{324 0 ^{\circ} n + 36 0 ^{\circ} + 54}{27}, x = 18 0^{\circ} - 4 0^{\circ} + 36 0^{\circ} n + 6

x = \frac{324 0 ^{\circ} n + 36 0 ^{\circ} + 54}{27}, x = 18 0^{\circ} - 4 0^{\circ} + 36 0^{\circ} n + 6

Popular Examples

sin(x)=0.9205

sin (x) = 0.9205

sin^2(x)+3sin(x)=-2,0<= x<= 2pi

sin^{2} (x) + 3 sin (x) = - 2, 0 \leq x \leq 2 π

solvefor x,-cos(x)=-sin^2(x)-1 solve for

x, - cos (x) = - sin^{2} (x) - 1

5sin(x)-2sin^2(x)-2=0

5 sin (x) - 2 sin^{2} (x) - 2 = 0

sin(x)-cos(x)= 7/5

sin (x) - cos (x) = \frac{7}{5}

Frequently Asked Questions (FAQ)

What is the general solution for cos(50+x)=sin(2x-6) ?
The general solution for cos(50+x)=sin(2x-6) is x=(3240n+360+54)/(27),x=180-40+360n+6