What is the general solution for sin(9x)-cos(x)=0 ?

The general solution for sin(9x)-cos(x)=0 is x=(pi+4pin)/(20),x=(pi+4pin)/(16)

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sin(9x)-cos(x)=0

Solution

sin (9 x) - cos (x) = 0

Solution

x = \frac{π + 4 πn}{20}, x = \frac{π + 4 πn}{16}

Degrees

x = 9^{\circ} + 3 6^{\circ} n, x = 11.2 5^{\circ} + 4 5^{\circ} n

Solution steps

sin (9 x) - cos (x) = 0

Add

cos (x)

to both sides

sin (9 x) = cos (x)

Rewrite using trig identities

sin (9 x) = cos (x)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (9 x) = sin (\frac{π}{2} - x)

sin (9 x) = sin (\frac{π}{2} - x)

Apply trig inverse properties

sin (9 x) = sin (\frac{π}{2} - x)

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

9 x = \frac{π}{2} - x + 2 πn, 9 x = π - (\frac{π}{2} - x) + 2 πn

9 x = \frac{π}{2} - x + 2 πn, 9 x = π - (\frac{π}{2} - x) + 2 πn

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

9 x = \frac{π}{2} - x + 2 πn

Move

x

to the left side

9 x = \frac{π}{2} - x + 2 πn

Add

x

to both sides

9 x + x = \frac{π}{2} - x + 2 πn + x

Simplify

10 x = \frac{π}{2} + 2 πn

10 x = \frac{π}{2} + 2 πn

Divide both sides by

10

10 x = \frac{π}{2} + 2 πn

Divide both sides by

10

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

Simplify

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

Simplify

\frac{10 x}{10} : x

\frac{10 x}{10}

Divide the numbers:

\frac{10}{10} = 1

= x

Simplify

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

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

Apply rule

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

= \frac{\frac{π}{2} + 2 πn}{10}

Join

\frac{π}{2} + 2 πn : \frac{π + 4 πn}{2}

\frac{π}{2} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}

= \frac{π}{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 πn \cdot 2}{2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π + 4 πn}{2}

= \frac{\frac{π + 4 πn}{2}}{10}

Apply the fraction rule:

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

= \frac{π + 4 πn}{2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{π + 4 πn}{20}

x = \frac{π + 4 πn}{20}

x = \frac{π + 4 πn}{20}

x = \frac{π + 4 πn}{20}

9 x = π - (\frac{π}{2} - x) + 2 πn : x = \frac{π + 4 πn}{16}

9 x = π - (\frac{π}{2} - x) + 2 πn

Expand

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

π - (\frac{π}{2} - x) + 2 πn

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

- (\frac{π}{2} - x)

Distribute parentheses

= - (\frac{π}{2}) - (- x)

Apply minus-plus rules

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

= - \frac{π}{2} + x

= π - \frac{π}{2} + x + 2 πn

9 x = π - \frac{π}{2} + x + 2 πn

Move

x

to the left side

9 x = π - \frac{π}{2} + x + 2 πn

Subtract

x

from both sides

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

Simplify

8 x = π - \frac{π}{2} + 2 πn

8 x = π - \frac{π}{2} + 2 πn

Divide both sides by

8

8 x = π - \frac{π}{2} + 2 πn

Divide both sides by

8

\frac{8 x}{8} = \frac{π}{8} - \frac{\frac{π}{2}}{8} + \frac{2 πn}{8}

Simplify

\frac{8 x}{8} = \frac{π}{8} - \frac{\frac{π}{2}}{8} + \frac{2 πn}{8}

Simplify

\frac{8 x}{8} : x

\frac{8 x}{8}

Divide the numbers:

\frac{8}{8} = 1

= x

Simplify

\frac{π}{8} - \frac{\frac{π}{2}}{8} + \frac{2 πn}{8} : \frac{π + 4 πn}{16}

\frac{π}{8} - \frac{\frac{π}{2}}{8} + \frac{2 πn}{8}

Apply rule

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

= \frac{π - \frac{π}{2} + 2 πn}{8}

Join

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

π - \frac{π}{2} + 2 πn

Convert element to fraction:

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

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

π 2 - π + 2 πn \cdot 2 = π + 4 πn

π 2 - π + 2 πn \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn

= \frac{π + 4 πn}{2}

= \frac{\frac{π + 4 πn}{2}}{8}

Apply the fraction rule:

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

= \frac{π + 4 πn}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{π + 4 πn}{16}

x = \frac{π + 4 πn}{16}

x = \frac{π + 4 πn}{16}

x = \frac{π + 4 πn}{16}

x = \frac{π + 4 πn}{20}, x = \frac{π + 4 πn}{16}

x = \frac{π + 4 πn}{20}, x = \frac{π + 4 πn}{16}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(9x)-cos(x)=0 ?
The general solution for sin(9x)-cos(x)=0 is x=(pi+4pin)/(20),x=(pi+4pin)/(16)