What is the general solution for sin(2x-10)=cos(x+40) ?

The general solution for sin(2x-10)=cos(x+40) is x=(1080n+180)/9

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Popular Trigonometry >

sin(2x-10)=cos(x+40)

Solution

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

Solution

x = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{9}

Radians

x = \frac{π}{9} + \frac{6 π}{9} n

Solution steps

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

Rewrite using trig identities

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

Use the following identity:

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

sin (2 x - 1 0^{\circ}) = sin (9 0^{\circ} - (x + 4 0^{\circ}))

sin (2 x - 1 0^{\circ}) = sin (9 0^{\circ} - (x + 4 0^{\circ}))

Apply trig inverse properties

sin (2 x - 1 0^{\circ}) = sin (9 0^{\circ} - (x + 4 0^{\circ}))

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

2 x - 1 0^{\circ} = 9 0^{\circ} - (x + 4 0^{\circ}) + 36 0^{\circ} n, 2 x - 1 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (x + 4 0^{\circ})) + 36 0^{\circ} n

2 x - 1 0^{\circ} = 9 0^{\circ} - (x + 4 0^{\circ}) + 36 0^{\circ} n, 2 x - 1 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (x + 4 0^{\circ})) + 36 0^{\circ} n

2 x - 1 0^{\circ} = 9 0^{\circ} - (x + 4 0^{\circ}) + 36 0^{\circ} n : x = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{9}

2 x - 1 0^{\circ} = 9 0^{\circ} - (x + 4 0^{\circ}) + 36 0^{\circ} n

Expand

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

9 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

= - x - 4 0^{\circ}

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

Simplify

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

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

Group like terms

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

Least Common Multiplier of

2, 9 : 18

2, 9

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

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

2

9

= 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}

For

4 0^{\circ} :

multiply the denominator and numerator by

2

4 0^{\circ} = \frac{36 0 ^{\circ} 2}{9 \cdot 2} = 4 0^{\circ}

= 9 0^{\circ} - 4 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 - 72 0 ^{\circ}}{18}

Add similar elements:

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

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

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

2 x - 1 0^{\circ} = - x + 36 0^{\circ} n + 5 0^{\circ}

Move

1 0^{\circ}

to the right side

2 x - 1 0^{\circ} = - x + 36 0^{\circ} n + 5 0^{\circ}

Add

1 0^{\circ}

to both sides

2 x - 1 0^{\circ} + 1 0^{\circ} = - x + 36 0^{\circ} n + 5 0^{\circ} + 1 0^{\circ}

Simplify

2 x - 1 0^{\circ} + 1 0^{\circ} = - x + 36 0^{\circ} n + 5 0^{\circ} + 1 0^{\circ}

Simplify

2 x - 1 0^{\circ} + 1 0^{\circ} : 2 x

2 x - 1 0^{\circ} + 1 0^{\circ}

Add similar elements:

- 1 0^{\circ} + 1 0^{\circ} = 0

= 2 x

Simplify

- x + 36 0^{\circ} n + 5 0^{\circ} + 1 0^{\circ} : - x + 36 0^{\circ} n + 6 0^{\circ}

- x + 36 0^{\circ} n + 5 0^{\circ} + 1 0^{\circ}

Combine the fractions

5 0^{\circ} + 1 0^{\circ} : 6 0^{\circ}

Apply rule

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

= \frac{90 0 ^{\circ} + 18 0 ^{\circ}}{18}

Add similar elements:

90 0^{\circ} + 18 0^{\circ} = 108 0^{\circ}

= 6 0^{\circ}

Cancel the common factor:

6

= 6 0^{\circ}

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

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

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

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

Move

x

to the left side

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

Add

x

to both sides

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

Simplify

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

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

Divide both sides by

3

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

Divide both sides by

3

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

Simplify

\frac{3 x}{3} = \frac{36 0 ^{\circ} n}{3} + \frac{6 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{6 0 ^{\circ}}{3} : \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{9}

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

Apply rule

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

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

Join

36 0^{\circ} n + 6 0^{\circ} : \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{3}

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

Convert element to fraction:

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

= \frac{36 0 ^{\circ} n \cdot 3}{3} + 6 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 3 + 18 0 ^{\circ}}{3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{3}

= \frac{\frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{3}}{3}

Apply the fraction rule:

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

= \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{9}

x = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{9}

x = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{9}

x = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{9}

2 x - 1 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (x + 4 0^{\circ})) + 36 0^{\circ} n : x = 18 0^{\circ} + 36 0^{\circ} n - 4 0^{\circ}

2 x - 1 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (x + 4 0^{\circ})) + 36 0^{\circ} n

Expand

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

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

Expand

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

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

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

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

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

= - x - 4 0^{\circ}

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

Simplify

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

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

Group like terms

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

Least Common Multiplier of

2, 9 : 18

2, 9

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

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

2

9

= 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}

For

4 0^{\circ} :

multiply the denominator and numerator by

2

4 0^{\circ} = \frac{36 0 ^{\circ} 2}{9 \cdot 2} = 4 0^{\circ}

= 9 0^{\circ} - 4 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 - 72 0 ^{\circ}}{18}

Add similar elements:

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

= - x + 5 0^{\circ}

= - x + 5 0^{\circ}

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

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

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

Distribute parentheses

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

Apply minus-plus rules

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

= x - 5 0^{\circ}

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

2 x - 1 0^{\circ} = 18 0^{\circ} + x - 5 0^{\circ} + 36 0^{\circ} n

Move

1 0^{\circ}

to the right side

2 x - 1 0^{\circ} = 18 0^{\circ} + x - 5 0^{\circ} + 36 0^{\circ} n

Add

1 0^{\circ}

to both sides

2 x - 1 0^{\circ} + 1 0^{\circ} = 18 0^{\circ} + x - 5 0^{\circ} + 36 0^{\circ} n + 1 0^{\circ}

Simplify

2 x - 1 0^{\circ} + 1 0^{\circ} = 18 0^{\circ} + x - 5 0^{\circ} + 36 0^{\circ} n + 1 0^{\circ}

Simplify

2 x - 1 0^{\circ} + 1 0^{\circ} : 2 x

2 x - 1 0^{\circ} + 1 0^{\circ}

Add similar elements:

- 1 0^{\circ} + 1 0^{\circ} = 0

= 2 x

Simplify

18 0^{\circ} + x - 5 0^{\circ} + 36 0^{\circ} n + 1 0^{\circ} : x + 18 0^{\circ} + 36 0^{\circ} n - 4 0^{\circ}

18 0^{\circ} + x - 5 0^{\circ} + 36 0^{\circ} n + 1 0^{\circ}

Group like terms

= x + 18 0^{\circ} + 36 0^{\circ} n + 1 0^{\circ} - 5 0^{\circ}

Combine the fractions

1 0^{\circ} - 5 0^{\circ} : - 4 0^{\circ}

Apply rule

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

= \frac{18 0 ^{\circ} - 90 0 ^{\circ}}{18}

Add similar elements:

18 0^{\circ} - 90 0^{\circ} = - 72 0^{\circ}

= \frac{- 72 0 ^{\circ}}{18}

Apply the fraction rule:

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

= - 4 0^{\circ}

Cancel the common factor:

2

= - 4 0^{\circ}

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

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

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

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

Move

x

to the left side

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

Subtract

x

from both sides

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

Simplify

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

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

Merge Overlapping Intervals

x = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{9}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(2x-10)=cos(x+40) ?
The general solution for sin(2x-10)=cos(x+40) is x=(1080n+180)/9