What is the general solution for 0= 20/3 sin(10t)+5cos(10t) ?

The general solution for 0= 20/3 sin(10t)+5cos(10t) is t=-(0.64350…)/(10)+(pin)/(10)

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Popular >

0= 20/3 sin(10t)+5cos(10t)

Solution

0 = \frac{20}{3} sin (10 t) + 5 cos (10 t)

Solution

t = - \frac{0.64350 \dots}{10} + \frac{πn}{10}

Degrees

t = - 3.68698 \dots^{\circ} + 1 8^{\circ} n

Solution steps

0 = \frac{20}{3} sin (10 t) + 5 cos (10 t)

Switch sides

\frac{20}{3} sin (10 t) + 5 cos (10 t) = 0

Rewrite using trig identities

\frac{20}{3} sin (10 t) + 5 cos (10 t) = 0

Divide both sides by

cos (10 t), cos (10 t) \neq = 0

\frac{\frac{20}{3} sin ( 10 t ) + 5 cos ( 10 t )}{cos ( 10 t )} = \frac{0}{cos ( 10 t )}

Simplify

\frac{20 sin ( 10 t )}{3 cos ( 10 t )} + 5 = 0

Use the basic trigonometric identity:

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

\frac{20 tan ( 10 t )}{3} + 5 = 0

\frac{20 tan ( 10 t )}{3} + 5 = 0

Move

5

to the right side

\frac{20 tan ( 10 t )}{3} + 5 = 0

Subtract

5

from both sides

\frac{20 tan ( 10 t )}{3} + 5 - 5 = 0 - 5

Simplify

\frac{20 tan ( 10 t )}{3} = - 5

\frac{20 tan ( 10 t )}{3} = - 5

Multiply both sides by

3

\frac{20 tan ( 10 t )}{3} = - 5

Multiply both sides by

3

\frac{3 \cdot 20 tan ( 10 t )}{3} = 3 (- 5)

Simplify

20 tan (10 t) = - 15

20 tan (10 t) = - 15

Divide both sides by

20

20 tan (10 t) = - 15

Divide both sides by

20

\frac{20 tan ( 10 t )}{20} = \frac{- 15}{20}

Simplify

\frac{20 tan ( 10 t )}{20} = \frac{- 15}{20}

Simplify

\frac{20 tan ( 10 t )}{20} : tan (10 t)

\frac{20 tan ( 10 t )}{20}

Divide the numbers:

\frac{20}{20} = 1

= tan (10 t)

Simplify

\frac{- 15}{20} : - \frac{3}{4}

\frac{- 15}{20}

Apply the fraction rule:

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

= - \frac{15}{20}

Cancel the common factor:

5

= - \frac{3}{4}

tan (10 t) = - \frac{3}{4}

tan (10 t) = - \frac{3}{4}

tan (10 t) = - \frac{3}{4}

Apply trig inverse properties

tan (10 t) = - \frac{3}{4}

General solutions for

tan (10 t) = - \frac{3}{4}

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

10 t = arctan (- \frac{3}{4}) + πn

10 t = arctan (- \frac{3}{4}) + πn

Solve

10 t = arctan (- \frac{3}{4}) + πn : t = - \frac{arctan ( \frac{3}{4} )}{10} + \frac{πn}{10}

10 t = arctan (- \frac{3}{4}) + πn

Simplify

arctan (- \frac{3}{4}) + πn : - arctan (\frac{3}{4}) + πn

arctan (- \frac{3}{4}) + πn

Use the following property:

arctan (- x) = - arctan (x)

arctan (- \frac{3}{4}) = - arctan (\frac{3}{4})

= - arctan (\frac{3}{4}) + πn

10 t = - arctan (\frac{3}{4}) + πn

Divide both sides by

10

10 t = - arctan (\frac{3}{4}) + πn

Divide both sides by

10

\frac{10 t}{10} = - \frac{arctan ( \frac{3}{4} )}{10} + \frac{πn}{10}

Simplify

t = - \frac{arctan ( \frac{3}{4} )}{10} + \frac{πn}{10}

t = - \frac{arctan ( \frac{3}{4} )}{10} + \frac{πn}{10}

t = - \frac{arctan ( \frac{3}{4} )}{10} + \frac{πn}{10}

Show solutions in decimal form

t = - \frac{0.64350 \dots}{10} + \frac{πn}{10}

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Frequently Asked Questions (FAQ)

What is the general solution for 0= 20/3 sin(10t)+5cos(10t) ?
The general solution for 0= 20/3 sin(10t)+5cos(10t) is t=-(0.64350…)/(10)+(pin)/(10)