What is the general solution for -cos(20.02t)=sin(20t) ?

The general solution for -cos(20.02t)=sin(20t) is t=25pi(4n+1),t=-(6.28318…n+pi/2)/(40.02)

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-cos(20.02t)=sin(20t)

Solution

- cos (20.02 t) = sin (20 t)

Solution

t = 25 π (4 n + 1), t = - \frac{6.28318 \dots n + \frac{π}{2}}{40.02}

Degrees

t = 450 0^{\circ} + 1800 0^{\circ} n, t = - 2.24887 \dots^{\circ} - 8.99550 \dots^{\circ} n

Solution steps

- cos (20.02 t) = sin (20 t)

Multiply by

- 1

cos (20.02 t) = - sin (20 t)

Rewrite using trig identities

cos (20.02 t) = - sin (20 t)

Use the following identity:

- sin (x) = sin (- x)

cos (20.02 t) = sin (- (20 t))

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (\frac{π}{2} - 20.02 t) = sin (- (20 t))

sin (\frac{π}{2} - 20.02 t) = sin (- (20 t))

Apply trig inverse properties

sin (\frac{π}{2} - 20.02 t) = sin (- (20 t))

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

- (20 t) = \frac{π}{2} - 20.02 t + 2 πn, - (20 t) = π - (\frac{π}{2} - 20.02 t) + 2 πn

- (20 t) = \frac{π}{2} - 20.02 t + 2 πn, - (20 t) = π - (\frac{π}{2} - 20.02 t) + 2 πn

- (20 t) = \frac{π}{2} - 20.02 t + 2 πn : t = 25 π (4 n + 1)

- (20 t) = \frac{π}{2} - 20.02 t + 2 πn

Multiply both sides by

100

- (20 t) = \frac{π}{2} - 20.02 t + 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

- 20 t \cdot 100 = \frac{π}{2} \cdot 100 - 20.02 t \cdot 100 + 2 πn \cdot 100

Refine

- 2000 t = 50 π - 2002 t + 200 πn

- 2000 t = 50 π - 2002 t + 200 πn

Move

2002 t

to the left side

- 2000 t = 50 π - 2002 t + 200 πn

Add

2002 t

to both sides

- 2000 t + 2002 t = 50 π - 2002 t + 200 πn + 2002 t

Simplify

2 t = 50 π + 200 πn

2 t = 50 π + 200 πn

Divide both sides by

2

2 t = 50 π + 200 πn

Divide both sides by

2

\frac{2 t}{2} = \frac{50 π}{2} + \frac{200 πn}{2}

Simplify

\frac{2 t}{2} = \frac{50 π}{2} + \frac{200 πn}{2}

Simplify

\frac{2 t}{2} : t

\frac{2 t}{2}

Divide the numbers:

\frac{2}{2} = 1

= t

Simplify

\frac{50 π}{2} + \frac{200 πn}{2} : 25 π (4 n + 1)

\frac{50 π}{2} + \frac{200 πn}{2}

Apply rule

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

= \frac{50 π + 200 πn}{2}

Factor

50 π + 200 πn : 50 π (1 + 4 n)

50 π + 200 πn

Rewrite as

= 1 \cdot 50 π + 4 \cdot 50 πn

Factor out common term

50 π

= 50 π (1 + 4 n)

= \frac{50 π ( 1 + 4 n )}{2}

Divide the numbers:

\frac{50}{2} = 25

= 25 π (4 n + 1)

t = 25 π (4 n + 1)

t = 25 π (4 n + 1)

t = 25 π (4 n + 1)

- (20 t) = π - (\frac{π}{2} - 20.02 t) + 2 πn : t = - \frac{6.28318 \dots n + \frac{π}{2}}{40.02}

- (20 t) = π - (\frac{π}{2} - 20.02 t) + 2 πn

Expand

- (20 t) : - 20 t

- (20 t)

Remove parentheses:

(a) = a

= - 20 t

Expand

π - (\frac{π}{2} - 20.02 t) + 2 πn : π - \frac{π}{2} + 20.02 t + 6.28318 \dots n

π - (\frac{π}{2} - 20.02 t) + 2 πn

Multiply the numbers:

2 \cdot 3.14159 \dots = 6.28318 \dots

= π - (- 20.02 t + \frac{π}{2}) + 6.28318 \dots n

- (\frac{π}{2} - 20.02 t) : - \frac{π}{2} + 20.02 t

- (\frac{π}{2} - 20.02 t)

Distribute parentheses

= - (\frac{π}{2}) - (- 20.02 t)

Apply minus-plus rules

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

= - \frac{π}{2} + 20.02 t

= π - \frac{π}{2} + 20.02 t + 6.28318 \dots n

- 20 t = π - \frac{π}{2} + 20.02 t + 6.28318 \dots n

Move

20.02 t

to the left side

- 20 t = π - \frac{π}{2} + 20.02 t + 6.28318 \dots n

Subtract

20.02 t

from both sides

- 20 t - 20.02 t = π - \frac{π}{2} + 20.02 t + 6.28318 \dots n - 20.02 t

Simplify

- 40.02 t = π - \frac{π}{2} + 6.28318 \dots n

- 40.02 t = π - \frac{π}{2} + 6.28318 \dots n

Divide both sides by

- 40.02

- 40.02 t = π - \frac{π}{2} + 6.28318 \dots n

Divide both sides by

- 40.02

\frac{- 40.02 t}{- 40.02} = \frac{π}{- 40.02} - \frac{\frac{π}{2}}{- 40.02} + \frac{6.28318 \dots n}{- 40.02}

Simplify

\frac{- 40.02 t}{- 40.02} = \frac{π}{- 40.02} - \frac{\frac{π}{2}}{- 40.02} + \frac{6.28318 \dots n}{- 40.02}

Simplify

\frac{- 40.02 t}{- 40.02} : t

\frac{- 40.02 t}{- 40.02}

Apply the fraction rule:

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

= \frac{40.02 t}{40.02}

Cancel the common factor:

40.02

= t

Simplify

\frac{π}{- 40.02} - \frac{\frac{π}{2}}{- 40.02} + \frac{6.28318 \dots n}{- 40.02} : - \frac{6.28318 \dots n + \frac{π}{2}}{40.02}

\frac{π}{- 40.02} - \frac{\frac{π}{2}}{- 40.02} + \frac{6.28318 \dots n}{- 40.02}

Apply rule

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

= \frac{π - \frac{π}{2} + 6.28318 \dots n}{- 40.02}

Apply the fraction rule:

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

= - \frac{π - \frac{π}{2} + 6.28318 \dots n}{40.02}

Join

π - \frac{π}{2} + 6.28318 \dots n : 6.28318 \dots n + \frac{π}{2}

π - \frac{π}{2} + 6.28318 \dots n

Convert element to fraction:

π = \frac{π 2}{2}

= \frac{π 2}{2} - \frac{π}{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}

Add similar elements:

2 π - π = π

= \frac{π}{2}

= - \frac{6.28318 \dots n + \frac{π}{2}}{40.02}

t = - \frac{6.28318 \dots n + \frac{π}{2}}{40.02}

t = - \frac{6.28318 \dots n + \frac{π}{2}}{40.02}

t = - \frac{6.28318 \dots n + \frac{π}{2}}{40.02}

t = 25 π (4 n + 1), t = - \frac{6.28318 \dots n + \frac{π}{2}}{40.02}

t = 25 π (4 n + 1), t = - \frac{6.28318 \dots n + \frac{π}{2}}{40.02}

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Frequently Asked Questions (FAQ)

What is the general solution for -cos(20.02t)=sin(20t) ?
The general solution for -cos(20.02t)=sin(20t) is t=25pi(4n+1),t=-(6.28318…n+pi/2)/(40.02)