What is the general solution for -csc^2(x)+2csc(x)cot(x)=0 ?

The general solution for -csc^2(x)+2csc(x)cot(x)=0 is x= pi/3+2pin,x=(5pi)/3+2pin

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Popular >

-csc^2(x)+2csc(x)cot(x)=0

Solution

- csc^{2} (x) + 2 csc (x) cot (x) = 0

Solution

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

- csc^{2} (x) + 2 csc (x) cot (x) = 0

Factor

- csc^{2} (x) + 2 csc (x) cot (x) : csc (x) (- csc (x) + 2 cot (x))

- csc^{2} (x) + 2 csc (x) cot (x)

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

csc^{2} (x) = csc (x) csc (x)

= - csc (x) csc (x) + 2 cot (x) csc (x)

Factor out common term

csc (x)

= csc (x) (- csc (x) + 2 cot (x))

csc (x) (- csc (x) + 2 cot (x)) = 0

Solving each part separately

csc (x) = 0 or - csc (x) + 2 cot (x) = 0

csc (x) = 0 :

No Solution

csc (x) = 0

csc (x) \leq - 1 or csc (x) \geq 1

No Solution

- csc (x) + 2 cot (x) = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

- csc (x) + 2 cot (x) = 0

Express with sin, cos

- csc (x) + 2 cot (x)

Use the basic trigonometric identity:

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

= - \frac{1}{sin ( x )} + 2 cot (x)

Use the basic trigonometric identity:

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

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

Simplify

- \frac{1}{sin ( x )} + 2 \cdot \frac{cos ( x )}{sin ( x )} : \frac{- 1 + 2 cos ( x )}{sin ( x )}

- \frac{1}{sin ( x )} + 2 \cdot \frac{cos ( x )}{sin ( x )}

Multiply

2 \cdot \frac{cos ( x )}{sin ( x )} : \frac{2 cos ( x )}{sin ( x )}

2 \cdot \frac{cos ( x )}{sin ( x )}

Multiply fractions:

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

= \frac{cos ( x ) \cdot 2}{sin ( x )}

= - \frac{1}{sin ( x )} + \frac{2 cos ( x )}{sin ( x )}

Apply rule

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

= \frac{- 1 + 2 cos ( x )}{sin ( x )}

= \frac{- 1 + 2 cos ( x )}{sin ( x )}

\frac{- 1 + 2 cos ( x )}{sin ( x )} = 0

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

- 1 + 2 cos (x) = 0

Move

1

to the right side

- 1 + 2 cos (x) = 0

Add

1

to both sides

- 1 + 2 cos (x) + 1 = 0 + 1

Simplify

2 cos (x) = 1

2 cos (x) = 1

Divide both sides by

2

2 cos (x) = 1

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{1}{2}

Simplify

cos (x) = \frac{1}{2}

cos (x) = \frac{1}{2}

General solutions for

cos (x) = \frac{1}{2}

cos (x)

periodicity table with

2 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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0=-6sin^2(θ)+3sin(θ)+2

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sin(2x+3/pi)= 1/2

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sqrt(3)cot(x/3)+1=0

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Frequently Asked Questions (FAQ)

What is the general solution for -csc^2(x)+2csc(x)cot(x)=0 ?
The general solution for -csc^2(x)+2csc(x)cot(x)=0 is x= pi/3+2pin,x=(5pi)/3+2pin