What is the general solution for 6cos(6t)+24sin(6t)=0 ?

The general solution for 6cos(6t)+24sin(6t)=0 is t=-(0.24497…)/6+(pin)/6

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Popular >

6cos(6t)+24sin(6t)=0

Solution

6 cos (6 t) + 24 sin (6 t) = 0

Solution

t = - \frac{0.24497 \dots}{6} + \frac{πn}{6}

Degrees

t = - 2.33937 \dots^{\circ} + 3 0^{\circ} n

Solution steps

6 cos (6 t) + 24 sin (6 t) = 0

Rewrite using trig identities

6 cos (6 t) + 24 sin (6 t) = 0

Divide both sides by

cos (6 t), cos (6 t) \neq = 0

\frac{6 cos ( 6 t ) + 24 sin ( 6 t )}{cos ( 6 t )} = \frac{0}{cos ( 6 t )}

Simplify

6 + \frac{24 sin ( 6 t )}{cos ( 6 t )} = 0

Use the basic trigonometric identity:

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

6 + 24 tan (6 t) = 0

6 + 24 tan (6 t) = 0

Move

6

to the right side

6 + 24 tan (6 t) = 0

Subtract

6

from both sides

6 + 24 tan (6 t) - 6 = 0 - 6

Simplify

24 tan (6 t) = - 6

24 tan (6 t) = - 6

Divide both sides by

24

24 tan (6 t) = - 6

Divide both sides by

24

\frac{24 tan ( 6 t )}{24} = \frac{- 6}{24}

Simplify

\frac{24 tan ( 6 t )}{24} = \frac{- 6}{24}

Simplify

\frac{24 tan ( 6 t )}{24} : tan (6 t)

\frac{24 tan ( 6 t )}{24}

Divide the numbers:

\frac{24}{24} = 1

= tan (6 t)

Simplify

\frac{- 6}{24} : - \frac{1}{4}

\frac{- 6}{24}

Apply the fraction rule:

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

= - \frac{6}{24}

Cancel the common factor:

6

= - \frac{1}{4}

tan (6 t) = - \frac{1}{4}

tan (6 t) = - \frac{1}{4}

tan (6 t) = - \frac{1}{4}

Apply trig inverse properties

tan (6 t) = - \frac{1}{4}

General solutions for

tan (6 t) = - \frac{1}{4}

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

6 t = arctan (- \frac{1}{4}) + πn

6 t = arctan (- \frac{1}{4}) + πn

Solve

6 t = arctan (- \frac{1}{4}) + πn : t = - \frac{arctan ( \frac{1}{4} )}{6} + \frac{πn}{6}

6 t = arctan (- \frac{1}{4}) + πn

Simplify

arctan (- \frac{1}{4}) + πn : - arctan (\frac{1}{4}) + πn

arctan (- \frac{1}{4}) + πn

Use the following property:

arctan (- x) = - arctan (x)

arctan (- \frac{1}{4}) = - arctan (\frac{1}{4})

= - arctan (\frac{1}{4}) + πn

6 t = - arctan (\frac{1}{4}) + πn

Divide both sides by

6

6 t = - arctan (\frac{1}{4}) + πn

Divide both sides by

6

\frac{6 t}{6} = - \frac{arctan ( \frac{1}{4} )}{6} + \frac{πn}{6}

Simplify

t = - \frac{arctan ( \frac{1}{4} )}{6} + \frac{πn}{6}

t = - \frac{arctan ( \frac{1}{4} )}{6} + \frac{πn}{6}

t = - \frac{arctan ( \frac{1}{4} )}{6} + \frac{πn}{6}

Show solutions in decimal form

t = - \frac{0.24497 \dots}{6} + \frac{πn}{6}

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Frequently Asked Questions (FAQ)

What is the general solution for 6cos(6t)+24sin(6t)=0 ?
The general solution for 6cos(6t)+24sin(6t)=0 is t=-(0.24497…)/6+(pin)/6