What is the general solution for sin(4x)=cos(3x+13) ?

The general solution for sin(4x)=cos(3x+13) is x=(pi-26+4pin)/(14),x=pi-pi/2+13+2pin

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sin(4x)=cos(3x+13)

Solution

sin (4 x) = cos (3 x + 13)

Solution

x = \frac{π - 26 + 4 πn}{14}, x = π - \frac{π}{2} + 13 + 2 πn

Degrees

x = - 93.54930 \dots^{\circ} + 51.42857 \dots^{\circ} n, x = 834.84513 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (4 x) = cos (3 x + 13)

Rewrite using trig identities

sin (4 x) = cos (3 x + 13)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (4 x) = sin (\frac{π}{2} - (3 x + 13))

sin (4 x) = sin (\frac{π}{2} - (3 x + 13))

Apply trig inverse properties

sin (4 x) = sin (\frac{π}{2} - (3 x + 13))

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

4 x = \frac{π}{2} - (3 x + 13) + 2 πn, 4 x = π - (\frac{π}{2} - (3 x + 13)) + 2 πn

4 x = \frac{π}{2} - (3 x + 13) + 2 πn, 4 x = π - (\frac{π}{2} - (3 x + 13)) + 2 πn

4 x = \frac{π}{2} - (3 x + 13) + 2 πn : x = \frac{π - 26 + 4 πn}{14}

4 x = \frac{π}{2} - (3 x + 13) + 2 πn

Expand

\frac{π}{2} - (3 x + 13) + 2 πn : \frac{π}{2} - 3 x - 13 + 2 πn

\frac{π}{2} - (3 x + 13) + 2 πn

- (3 x + 13) : - 3 x - 13

- (3 x + 13)

Distribute parentheses

= - (3 x) - (13)

Apply minus-plus rules

+ (- a) = - a

= - 3 x - 13

= \frac{π}{2} - 3 x - 13 + 2 πn

4 x = \frac{π}{2} - 3 x - 13 + 2 πn

Move

3 x

to the left side

4 x = \frac{π}{2} - 3 x - 13 + 2 πn

Add

3 x

to both sides

4 x + 3 x = \frac{π}{2} - 3 x - 13 + 2 πn + 3 x

Simplify

7 x = \frac{π}{2} - 13 + 2 πn

7 x = \frac{π}{2} - 13 + 2 πn

Divide both sides by

7

7 x = \frac{π}{2} - 13 + 2 πn

Divide both sides by

7

\frac{7 x}{7} = \frac{\frac{π}{2}}{7} - \frac{13}{7} + \frac{2 πn}{7}

Simplify

\frac{7 x}{7} = \frac{\frac{π}{2}}{7} - \frac{13}{7} + \frac{2 πn}{7}

Simplify

\frac{7 x}{7} : x

\frac{7 x}{7}

Divide the numbers:

\frac{7}{7} = 1

= x

Simplify

\frac{\frac{π}{2}}{7} - \frac{13}{7} + \frac{2 πn}{7} : \frac{π - 26 + 4 πn}{14}

\frac{\frac{π}{2}}{7} - \frac{13}{7} + \frac{2 πn}{7}

Apply rule

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

= \frac{\frac{π}{2} - 13 + 2 πn}{7}

Join

\frac{π}{2} - 13 + 2 πn : \frac{π - 26 + 4 πn}{2}

\frac{π}{2} - 13 + 2 πn

Convert element to fraction:

13 = \frac{13 \cdot 2}{2}, 2 πn = \frac{2 πn 2}{2}

= \frac{π}{2} - \frac{13 \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 - 13 \cdot 2 + 2 πn \cdot 2}{2}

π - 13 \cdot 2 + 2 πn \cdot 2 = π - 26 + 4 πn

π - 13 \cdot 2 + 2 πn \cdot 2

Multiply the numbers:

13 \cdot 2 = 26

= π - 26 + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= π - 26 + 4 πn

= \frac{π - 26 + 4 πn}{2}

= \frac{\frac{π - 26 + 4 πn}{2}}{7}

Apply the fraction rule:

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

= \frac{π - 26 + 4 πn}{2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{π - 26 + 4 πn}{14}

x = \frac{π - 26 + 4 πn}{14}

x = \frac{π - 26 + 4 πn}{14}

x = \frac{π - 26 + 4 πn}{14}

4 x = π - (\frac{π}{2} - (3 x + 13)) + 2 πn : x = π - \frac{π}{2} + 13 + 2 πn

4 x = π - (\frac{π}{2} - (3 x + 13)) + 2 πn

Expand

π - (\frac{π}{2} - (3 x + 13)) + 2 πn : π - \frac{π}{2} + 3 x + 13 + 2 πn

π - (\frac{π}{2} - (3 x + 13)) + 2 πn

- (3 x + 13) : - 3 x - 13

- (3 x + 13)

Distribute parentheses

= - (3 x) - (13)

Apply minus-plus rules

+ (- a) = - a

= - 3 x - 13

= π - (- 3 x + \frac{π}{2} - 13) + 2 πn

- (\frac{π}{2} - 3 x - 13) : - \frac{π}{2} + 3 x + 13

- (\frac{π}{2} - 3 x - 13)

Distribute parentheses

= - (\frac{π}{2}) - (- 3 x) - (- 13)

Apply minus-plus rules

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

= - \frac{π}{2} + 3 x + 13

= π - \frac{π}{2} + 3 x + 13 + 2 πn

4 x = π - \frac{π}{2} + 3 x + 13 + 2 πn

Move

3 x

to the left side

4 x = π - \frac{π}{2} + 3 x + 13 + 2 πn

Subtract

3 x

from both sides

4 x - 3 x = π - \frac{π}{2} + 3 x + 13 + 2 πn - 3 x

Simplify

x = π - \frac{π}{2} + 13 + 2 πn

x = π - \frac{π}{2} + 13 + 2 πn

x = \frac{π - 26 + 4 πn}{14}, x = π - \frac{π}{2} + 13 + 2 πn

x = \frac{π - 26 + 4 πn}{14}, x = π - \frac{π}{2} + 13 + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin(4x)=cos(3x+13) ?
The general solution for sin(4x)=cos(3x+13) is x=(pi-26+4pin)/(14),x=pi-pi/2+13+2pin