What is the general solution for 0=(6cos(30t)+2sin(30t)) ?

The general solution for 0=(6cos(30t)+2sin(30t)) is t=-(1.24904…)/(30)+(pin)/(30)

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Popular >

0=(6cos(30t)+2sin(30t))

Solution

0 = (6 cos (30 t) + 2 sin (30 t))

Solution

t = - \frac{1.24904 \dots}{30} + \frac{πn}{30}

Degrees

t = - 2.38550 \dots^{\circ} + 6^{\circ} n

Solution steps

0 = (6 cos (30 t) + 2 sin (30 t))

Switch sides

(6 cos (30 t) + 2 sin (30 t)) = 0

Rewrite using trig identities

(6 cos (30 t) + 2 sin (30 t)) = 0

Divide both sides by

cos (30 t), cos (30 t) \neq = 0

\frac{( 6 cos ( 30 t ) + 2 sin ( 30 t ) )}{cos ( 30 t )} = \frac{0}{cos ( 30 t )}

Simplify

6 + \frac{2 sin ( 30 t )}{cos ( 30 t )} = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

6 + 2 tan (30 t) = 0

6 + 2 tan (30 t) = 0

Move

6

to the right side

6 + 2 tan (30 t) = 0

Subtract

6

from both sides

6 + 2 tan (30 t) - 6 = 0 - 6

Simplify

2 tan (30 t) = - 6

2 tan (30 t) = - 6

Divide both sides by

2

2 tan (30 t) = - 6

Divide both sides by

2

\frac{2 tan ( 30 t )}{2} = \frac{- 6}{2}

Simplify

\frac{2 tan ( 30 t )}{2} = \frac{- 6}{2}

Simplify

\frac{2 tan ( 30 t )}{2} : tan (30 t)

\frac{2 tan ( 30 t )}{2}

Divide the numbers:

\frac{2}{2} = 1

= tan (30 t)

Simplify

\frac{- 6}{2} : - 3

\frac{- 6}{2}

Apply the fraction rule:

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

= - \frac{6}{2}

Divide the numbers:

\frac{6}{2} = 3

= - 3

tan (30 t) = - 3

tan (30 t) = - 3

tan (30 t) = - 3

Apply trig inverse properties

tan (30 t) = - 3

General solutions for

tan (30 t) = - 3

tan (x) = - a \Rightarrow x = arctan (- a) + πn

30 t = arctan (- 3) + πn

30 t = arctan (- 3) + πn

Solve

30 t = arctan (- 3) + πn : t = - \frac{arctan ( 3 )}{30} + \frac{πn}{30}

30 t = arctan (- 3) + πn

Simplify

arctan (- 3) + πn : - arctan (3) + πn

arctan (- 3) + πn

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 3) = - arctan (3)

= - arctan (3) + πn

30 t = - arctan (3) + πn

Divide both sides by

30

30 t = - arctan (3) + πn

Divide both sides by

30

\frac{30 t}{30} = - \frac{arctan ( 3 )}{30} + \frac{πn}{30}

Simplify

t = - \frac{arctan ( 3 )}{30} + \frac{πn}{30}

t = - \frac{arctan ( 3 )}{30} + \frac{πn}{30}

t = - \frac{arctan ( 3 )}{30} + \frac{πn}{30}

Show solutions in decimal form

t = - \frac{1.24904 \dots}{30} + \frac{πn}{30}

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Frequently Asked Questions (FAQ)

What is the general solution for 0=(6cos(30t)+2sin(30t)) ?
The general solution for 0=(6cos(30t)+2sin(30t)) is t=-(1.24904…)/(30)+(pin)/(30)