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

The general solution for sin(x)=cos(2x+10) is x=(pi-20+4pin)/6 ,x=-(pi+20+4pin)/2

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

Solution

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

Solution

x = \frac{π - 20 + 4 πn}{6}, x = - \frac{π + 20 + 4 πn}{2}

Degrees

x = - 160.98593 \dots^{\circ} + 12 0^{\circ} n, x = - 662.95779 \dots^{\circ} - 36 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (x) = sin (\frac{π}{2} - (2 x + 10))

sin (x) = sin (\frac{π}{2} - (2 x + 10))

Apply trig inverse properties

sin (x) = sin (\frac{π}{2} - (2 x + 10))

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

x = \frac{π}{2} - (2 x + 10) + 2 πn, x = π - (\frac{π}{2} - (2 x + 10)) + 2 πn

x = \frac{π}{2} - (2 x + 10) + 2 πn, x = π - (\frac{π}{2} - (2 x + 10)) + 2 πn

x = \frac{π}{2} - (2 x + 10) + 2 πn : x = \frac{π - 20 + 4 πn}{6}

x = \frac{π}{2} - (2 x + 10) + 2 πn

Expand

\frac{π}{2} - (2 x + 10) + 2 πn : \frac{π}{2} - 2 x - 10 + 2 πn

\frac{π}{2} - (2 x + 10) + 2 πn

- (2 x + 10) : - 2 x - 10

- (2 x + 10)

Distribute parentheses

= - (2 x) - (10)

Apply minus-plus rules

+ (- a) = - a

= - 2 x - 10

= \frac{π}{2} - 2 x - 10 + 2 πn

x = \frac{π}{2} - 2 x - 10 + 2 πn

Move

2 x

to the left side

x = \frac{π}{2} - 2 x - 10 + 2 πn

Add

2 x

to both sides

x + 2 x = \frac{π}{2} - 2 x - 10 + 2 πn + 2 x

Simplify

3 x = \frac{π}{2} - 10 + 2 πn

3 x = \frac{π}{2} - 10 + 2 πn

Divide both sides by

3

3 x = \frac{π}{2} - 10 + 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{\frac{π}{2}}{3} - \frac{10}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} = \frac{\frac{π}{2}}{3} - \frac{10}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{π}{2}}{3} - \frac{10}{3} + \frac{2 πn}{3} : \frac{π - 20 + 4 πn}{6}

\frac{\frac{π}{2}}{3} - \frac{10}{3} + \frac{2 πn}{3}

Apply rule

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

= \frac{\frac{π}{2} - 10 + 2 πn}{3}

Join

\frac{π}{2} - 10 + 2 πn : \frac{π - 20 + 4 πn}{2}

\frac{π}{2} - 10 + 2 πn

Convert element to fraction:

10 = \frac{10 \cdot 2}{2}, 2 πn = \frac{2 πn 2}{2}

= \frac{π}{2} - \frac{10 \cdot 2}{2} + \frac{2 πn \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{π - 10 \cdot 2 + 2 πn \cdot 2}{2}

π - 10 \cdot 2 + 2 πn \cdot 2 = π - 20 + 4 πn

π - 10 \cdot 2 + 2 πn \cdot 2

Multiply the numbers:

10 \cdot 2 = 20

= π - 20 + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= π - 20 + 4 πn

= \frac{π - 20 + 4 πn}{2}

= \frac{\frac{π - 20 + 4 πn}{2}}{3}

Apply the fraction rule:

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

= \frac{π - 20 + 4 πn}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{π - 20 + 4 πn}{6}

x = \frac{π - 20 + 4 πn}{6}

x = \frac{π - 20 + 4 πn}{6}

x = \frac{π - 20 + 4 πn}{6}

x = π - (\frac{π}{2} - (2 x + 10)) + 2 πn : x = - \frac{π + 20 + 4 πn}{2}

x = π - (\frac{π}{2} - (2 x + 10)) + 2 πn

Expand

π - (\frac{π}{2} - (2 x + 10)) + 2 πn : π - \frac{π}{2} + 2 x + 10 + 2 πn

π - (\frac{π}{2} - (2 x + 10)) + 2 πn

- (2 x + 10) : - 2 x - 10

- (2 x + 10)

Distribute parentheses

= - (2 x) - (10)

Apply minus-plus rules

+ (- a) = - a

= - 2 x - 10

= π - (- 2 x + \frac{π}{2} - 10) + 2 πn

- (\frac{π}{2} - 2 x - 10) : - \frac{π}{2} + 2 x + 10

- (\frac{π}{2} - 2 x - 10)

Distribute parentheses

= - (\frac{π}{2}) - (- 2 x) - (- 10)

Apply minus-plus rules

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

= - \frac{π}{2} + 2 x + 10

= π - \frac{π}{2} + 2 x + 10 + 2 πn

x = π - \frac{π}{2} + 2 x + 10 + 2 πn

Move

2 x

to the left side

x = π - \frac{π}{2} + 2 x + 10 + 2 πn

Subtract

2 x

from both sides

x - 2 x = π - \frac{π}{2} + 2 x + 10 + 2 πn - 2 x

Simplify

- x = π - \frac{π}{2} + 10 + 2 πn

- x = π - \frac{π}{2} + 10 + 2 πn

Divide both sides by

- 1

- x = π - \frac{π}{2} + 10 + 2 πn

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{π}{- 1} - \frac{\frac{π}{2}}{- 1} + \frac{10}{- 1} + \frac{2 πn}{- 1}

Simplify

\frac{- x}{- 1} = \frac{π}{- 1} - \frac{\frac{π}{2}}{- 1} + \frac{10}{- 1} + \frac{2 πn}{- 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{π}{- 1} - \frac{\frac{π}{2}}{- 1} + \frac{10}{- 1} + \frac{2 πn}{- 1} : - \frac{π + 20 + 4 πn}{2}

\frac{π}{- 1} - \frac{\frac{π}{2}}{- 1} + \frac{10}{- 1} + \frac{2 πn}{- 1}

Apply rule

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

= \frac{π - \frac{π}{2} + 10 + 2 πn}{- 1}

Apply the fraction rule:

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

= - \frac{π - \frac{π}{2} + 10 + 2 πn}{1}

Join

π - \frac{π}{2} + 10 + 2 πn : \frac{π + 20 + 4 πn}{2}

π - \frac{π}{2} + 10 + 2 πn

Convert element to fraction:

π = \frac{π 2}{2}, 10 = \frac{10 \cdot 2}{2}, 2 πn = \frac{2 πn 2}{2}

= \frac{π 2}{2} - \frac{π}{2} + \frac{10 \cdot 2}{2} + \frac{2 πn \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 - π + 10 \cdot 2 + 2 πn \cdot 2}{2}

π 2 - π + 10 \cdot 2 + 2 πn \cdot 2 = π + 20 + 4 πn

π 2 - π + 10 \cdot 2 + 2 πn \cdot 2

Add similar elements:

2 π - π = π

= π + 10 \cdot 2 + 2 \cdot 2 πn

Multiply the numbers:

10 \cdot 2 = 20

= π + 20 + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= π + 20 + 4 πn

= \frac{π + 20 + 4 πn}{2}

= - \frac{\frac{π + 4 πn + 20}{2}}{1}

Apply the fraction rule:

\frac{a}{1} = a

= - \frac{π + 4 πn + 20}{2}

x = - \frac{π + 20 + 4 πn}{2}

x = - \frac{π + 20 + 4 πn}{2}

x = - \frac{π + 20 + 4 πn}{2}

x = \frac{π - 20 + 4 πn}{6}, x = - \frac{π + 20 + 4 πn}{2}

x = \frac{π - 20 + 4 πn}{6}, x = - \frac{π + 20 + 4 πn}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)=cos(2x+10) ?
The general solution for sin(x)=cos(2x+10) is x=(pi-20+4pin)/6 ,x=-(pi+20+4pin)/2