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

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

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Popular >

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

Solution

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

Solution

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

Degrees

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

Solution steps

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

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

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

Move

5

to the right side

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

Subtract

5

from both sides

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

Simplify

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

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

Multiply both sides by

3

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

Multiply both sides by

3

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

Simplify

20 tan (10 x) = - 15

20 tan (10 x) = - 15

Divide both sides by

20

20 tan (10 x) = - 15

Divide both sides by

20

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

Simplify

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

Simplify

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

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

Divide the numbers:

\frac{20}{20} = 1

= tan (10 x)

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 x) = - \frac{3}{4}

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

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

Apply trig inverse properties

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

General solutions for

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

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

10 x = arctan (- \frac{3}{4}) + πn

10 x = arctan (- \frac{3}{4}) + πn

Solve

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

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

Divide both sides by

10

10 x = - arctan (\frac{3}{4}) + πn

Divide both sides by

10

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

Simplify

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

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

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

Show solutions in decimal form

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

Popular Examples

5sin^2(x)=0

5 sin^{2} (x) = 0

(((1-sin^2(x)))/((1+2*sin^2(x))))=0.4

(\frac{( 1 - sin ^{2} ( x ) )}{( 1 + 2 \cdot sin ^{2} ( x ) )}) = 0.4

cos(3x+pi/4)=-(sqrt(3))/2

cos (3 x + \frac{π}{4}) = - \frac{3}{2}

1=cos(pi/(64)*t^2)

1 = cos (\frac{π}{64} \cdot t^{2})

sin(x)-1/4 =0

sin (x) - \frac{1}{4} = 0

Frequently Asked Questions (FAQ)

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