What is the general solution for 5/(csc(x))-5sqrt(3)cos(x)=0 ?

The general solution for 5/(csc(x))-5sqrt(3)cos(x)=0 is x= pi/3+pin

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Popular >

5/(csc(x))-5sqrt(3)cos(x)=0

Solution

\frac{5}{csc ( x )} - 53 cos (x) = 0

Solution

x = \frac{π}{3} + πn

Degrees

x = 6 0^{\circ} + 18 0^{\circ} n

Solution steps

\frac{5}{csc ( x )} - 53 cos (x) = 0

Simplify

\frac{5}{csc ( x )} - 53 cos (x) : \frac{5 - 5 3 cos ( x ) csc ( x )}{csc ( x )}

\frac{5}{csc ( x )} - 53 cos (x)

Convert element to fraction:

53 cos (x) = \frac{5 \cdot 3 cos ( x ) csc ( x )}{csc ( x )}

= \frac{5}{csc ( x )} - \frac{5 3 cos ( x ) csc ( x )}{csc ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{5 - 5 3 cos ( x ) csc ( x )}{csc ( x )}

\frac{5 - 5 3 cos ( x ) csc ( x )}{csc ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

5 - 53 cos (x) csc (x) = 0

Rewrite using trig identities

5 - 53 cos (x) csc (x)

csc (x) cos (x) = cot (x)

csc (x) cos (x)

Express with sin, cos

csc (x) cos (x)

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

= \frac{1}{sin ( x )} cos (x)

Simplify

\frac{1}{sin ( x )} cos (x) : \frac{cos ( x )}{sin ( x )}

\frac{1}{sin ( x )} cos (x)

Multiply fractions:

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

= \frac{1 cos ( x )}{sin ( x )}

Multiply:

1 \cdot cos (x) = cos (x)

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

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

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

Use the basic trigonometric identity:

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

= cot (x)

= 5 - 53 cot (x)

5 - 53 cot (x) = 0

Move

5

to the right side

5 - 53 cot (x) = 0

Subtract

5

from both sides

5 - 53 cot (x) - 5 = 0 - 5

Simplify

- 53 cot (x) = - 5

- 53 cot (x) = - 5

Divide both sides by

- 53

- 53 cot (x) = - 5

Divide both sides by

- 53

\frac{- 5 3 cot ( x )}{- 5 3} = \frac{- 5}{- 5 3}

Simplify

\frac{- 5 3 cot ( x )}{- 5 3} = \frac{- 5}{- 5 3}

Simplify

\frac{- 5 3 cot ( x )}{- 5 3} : cot (x)

\frac{- 5 3 cot ( x )}{- 5 3}

Apply the fraction rule:

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

= \frac{5 3 cot ( x )}{5 3}

Divide the numbers:

\frac{5}{5} = 1

= \frac{3 cot ( x )}{3}

Cancel the common factor:

3

= cot (x)

Simplify

\frac{- 5}{- 5 3} : \frac{3}{3}

\frac{- 5}{- 5 3}

Apply the fraction rule:

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

= \frac{5}{5 3}

Divide the numbers:

\frac{5}{5} = 1

= \frac{1}{3}

Rationalize

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

= \frac{3}{3}

cot (x) = \frac{3}{3}

cot (x) = \frac{3}{3}

cot (x) = \frac{3}{3}

General solutions for

cot (x) = \frac{3}{3}

cot (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{3} + πn

x = \frac{π}{3} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 5/(csc(x))-5sqrt(3)cos(x)=0 ?
The general solution for 5/(csc(x))-5sqrt(3)cos(x)=0 is x= pi/3+pin