What is the general solution for csc(x)-cot(x)=3 ?

The general solution for csc(x)-cot(x)=3 is x=2.49809…+2pin

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csc(x)-cot(x)=3

Solution

csc (x) - cot (x) = 3

Solution

x = 2.49809 \dots + 2 πn

Degrees

x = 143.13010 \dots^{\circ} + 36 0^{\circ} n

Solution steps

csc (x) - cot (x) = 3

Subtract

3

from both sides

csc (x) - cot (x) - 3 = 0

Express with sin, cos

\frac{1}{sin ( x )} - \frac{cos ( x )}{sin ( x )} - 3 = 0

Simplify

\frac{1}{sin ( x )} - \frac{cos ( x )}{sin ( x )} - 3 : \frac{1 - cos ( x ) - 3 sin ( x )}{sin ( x )}

\frac{1}{sin ( x )} - \frac{cos ( x )}{sin ( x )} - 3

Combine the fractions

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

Apply rule

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

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

= \frac{- cos ( x ) + 1}{sin ( x )} - 3

Convert element to fraction:

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

= \frac{1 - cos ( x )}{sin ( x )} - \frac{3 sin ( x )}{sin ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{1 - cos ( x ) - 3 sin ( x )}{sin ( x )}

\frac{1 - cos ( x ) - 3 sin ( x )}{sin ( x )} = 0

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

1 - cos (x) - 3 sin (x) = 0

Add

3 sin (x)

to both sides

1 - cos (x) = 3 sin (x)

Square both sides

(1 - cos (x))^{2} = (3 sin (x))^{2}

Subtract

(3 sin (x))^{2}

from both sides

(1 - cos (x))^{2} - 9 sin^{2} (x) = 0

Rewrite using trig identities

(1 - cos (x))^{2} - 9 sin^{2} (x)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= (1 - cos (x))^{2} - 9 (1 - cos^{2} (x))

Simplify

(1 - cos (x))^{2} - 9 (1 - cos^{2} (x)) : 10 cos^{2} (x) - 2 cos (x) - 8

(1 - cos (x))^{2} - 9 (1 - cos^{2} (x))

(1 - cos (x))^{2} : 1 - 2 cos (x) + cos^{2} (x)

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 1, b = cos (x)

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

Simplify

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

Multiply the numbers:

2 \cdot 1 = 2

= 1 - 2 cos (x) + cos^{2} (x)

= 1 - 2 cos (x) + cos^{2} (x)

= 1 - 2 cos (x) + cos^{2} (x) - 9 (1 - cos^{2} (x))

Expand

- 9 (1 - cos^{2} (x)) : - 9 + 9 cos^{2} (x)

- 9 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 9, b = 1, c = cos^{2} (x)

= - 9 \cdot 1 - (- 9) cos^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 9 \cdot 1 + 9 cos^{2} (x)

Multiply the numbers:

9 \cdot 1 = 9

= - 9 + 9 cos^{2} (x)

= 1 - 2 cos (x) + cos^{2} (x) - 9 + 9 cos^{2} (x)

Simplify

1 - 2 cos (x) + cos^{2} (x) - 9 + 9 cos^{2} (x) : 10 cos^{2} (x) - 2 cos (x) - 8

1 - 2 cos (x) + cos^{2} (x) - 9 + 9 cos^{2} (x)

Group like terms

= - 2 cos (x) + cos^{2} (x) + 9 cos^{2} (x) + 1 - 9

Add similar elements:

cos^{2} (x) + 9 cos^{2} (x) = 10 cos^{2} (x)

= - 2 cos (x) + 10 cos^{2} (x) + 1 - 9

Add/Subtract the numbers:

1 - 9 = - 8

= 10 cos^{2} (x) - 2 cos (x) - 8

= 10 cos^{2} (x) - 2 cos (x) - 8

= 10 cos^{2} (x) - 2 cos (x) - 8

- 8 + 10 cos^{2} (x) - 2 cos (x) = 0

Solve by substitution

- 8 + 10 cos^{2} (x) - 2 cos (x) = 0

Let:

cos (x) = u

- 8 + 10 u^{2} - 2 u = 0

- 8 + 10 u^{2} - 2 u = 0 : u = 1, u = - \frac{4}{5}

- 8 + 10 u^{2} - 2 u = 0

Write in the standard form

a x^{2} + b x + c = 0

10 u^{2} - 2 u - 8 = 0

Solve with the quadratic formula

10 u^{2} - 2 u - 8 = 0

Quadratic Equation Formula:

For

a = 10, b = - 2, c = - 8

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 10 ( - 8 )}{2 \cdot 10}

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 10 ( - 8 )}{2 \cdot 10}

(- 2)^{2} - 4 \cdot 10 (- 8) = 18

(- 2)^{2} - 4 \cdot 10 (- 8)

Apply rule

- (- a) = a

= (- 2)^{2} + 4 \cdot 10 \cdot 8

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 2)^{2} = 2^{2}

= 2^{2} + 4 \cdot 10 \cdot 8

Multiply the numbers:

4 \cdot 10 \cdot 8 = 320

= 2^{2} + 320

2^{2} = 4

= 4 + 320

Add the numbers:

4 + 320 = 324

= 324

Factor the number:

324 = 1 8^{2}

= 1 8^{2}

Apply radical rule:

1 8^{2} = 18

= 18

u_{1, 2} = \frac{- ( - 2 ) \pm 18}{2 \cdot 10}

Separate the solutions

u_{1} = \frac{- ( - 2 ) + 18}{2 \cdot 10}, u_{2} = \frac{- ( - 2 ) - 18}{2 \cdot 10}

u = \frac{- ( - 2 ) + 18}{2 \cdot 10} : 1

\frac{- ( - 2 ) + 18}{2 \cdot 10}

Apply rule

- (- a) = a

= \frac{2 + 18}{2 \cdot 10}

Add the numbers:

2 + 18 = 20

= \frac{20}{2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{20}{20}

Apply rule

\frac{a}{a} = 1

= 1

u = \frac{- ( - 2 ) - 18}{2 \cdot 10} : - \frac{4}{5}

\frac{- ( - 2 ) - 18}{2 \cdot 10}

Apply rule

- (- a) = a

= \frac{2 - 18}{2 \cdot 10}

Subtract the numbers:

2 - 18 = - 16

= \frac{- 16}{2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{- 16}{20}

Apply the fraction rule:

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

= - \frac{16}{20}

Cancel the common factor:

4

= - \frac{4}{5}

The solutions to the quadratic equation are:

u = 1, u = - \frac{4}{5}

Substitute back

u = cos (x)

cos (x) = 1, cos (x) = - \frac{4}{5}

cos (x) = 1, cos (x) = - \frac{4}{5}

cos (x) = 1 : x = 2 πn

cos (x) = 1

General solutions for

cos (x) = 1

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 = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

cos (x) = - \frac{4}{5} : x = arccos (- \frac{4}{5}) + 2 πn, x = - arccos (- \frac{4}{5}) + 2 πn

cos (x) = - \frac{4}{5}

Apply trig inverse properties

cos (x) = - \frac{4}{5}

General solutions for

cos (x) = - \frac{4}{5}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{4}{5}) + 2 πn, x = - arccos (- \frac{4}{5}) + 2 πn

x = arccos (- \frac{4}{5}) + 2 πn, x = - arccos (- \frac{4}{5}) + 2 πn

Combine all the solutions

x = 2 πn, x = arccos (- \frac{4}{5}) + 2 πn, x = - arccos (- \frac{4}{5}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

csc (x) - cot (x) = 3

Remove the ones that don't agree with the equation.

Check the solution

2 πn :

False

2 πn

Plug in

n = 1

2 π 1

For

csc (x) - cot (x) = 3

plug in

x = 2 π 1

csc (2 π 1) - cot (2 π 1) = 3

Undefined

\Rightarrow False

Check the solution

arccos (- \frac{4}{5}) + 2 πn :

True

arccos (- \frac{4}{5}) + 2 πn

Plug in

n = 1

arccos (- \frac{4}{5}) + 2 π 1

For

csc (x) - cot (x) = 3

plug in

x = arccos (- \frac{4}{5}) + 2 π 1

csc (arccos (- \frac{4}{5}) + 2 π 1) - cot (arccos (- \frac{4}{5}) + 2 π 1) = 3

Refine

3 = 3

\Rightarrow True

Check the solution

- arccos (- \frac{4}{5}) + 2 πn :

False

- arccos (- \frac{4}{5}) + 2 πn

Plug in

n = 1

- arccos (- \frac{4}{5}) + 2 π 1

For

csc (x) - cot (x) = 3

plug in

x = - arccos (- \frac{4}{5}) + 2 π 1

csc (- arccos (- \frac{4}{5}) + 2 π 1) - cot (- arccos (- \frac{4}{5}) + 2 π 1) = 3

Refine

- 3 = 3

\Rightarrow False

x = arccos (- \frac{4}{5}) + 2 πn

Show solutions in decimal form

x = 2.49809 \dots + 2 πn

Popular Examples

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cos (θ) = \frac{5}{20}

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Frequently Asked Questions (FAQ)

What is the general solution for csc(x)-cot(x)=3 ?
The general solution for csc(x)-cot(x)=3 is x=2.49809…+2pin