What is the general solution for csc(X)-cot(X)=5 ?

The general solution for csc(X)-cot(X)=5 is X=2.74680…+2pin

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Popular Trigonometry >

csc(X)-cot(X)=5

Solution

csc (X) - cot (X) = 5

Solution

X = 2.74680 \dots + 2 πn

Degrees

X = 157.38013 \dots^{\circ} + 36 0^{\circ} n

Solution steps

csc (X) - cot (X) = 5

Subtract

5

from both sides

csc (X) - cot (X) - 5 = 0

Express with sin, cos

\frac{1}{sin ( X )} - \frac{cos ( X )}{sin ( X )} - 5 = 0

Simplify

\frac{1}{sin ( X )} - \frac{cos ( X )}{sin ( X )} - 5 : \frac{1 - cos ( X ) - 5 sin ( X )}{sin ( X )}

\frac{1}{sin ( X )} - \frac{cos ( X )}{sin ( X )} - 5

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 )} - 5

Convert element to fraction:

5 = \frac{5 sin ( X )}{sin ( X )}

= \frac{1 - cos ( X )}{sin ( X )} - \frac{5 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 ) - 5 sin ( X )}{sin ( X )}

\frac{1 - cos ( X ) - 5 sin ( X )}{sin ( X )} = 0

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

1 - cos (X) - 5 sin (X) = 0

Add

5 sin (X)

to both sides

1 - cos (X) = 5 sin (X)

Square both sides

(1 - cos (X))^{2} = (5 sin (X))^{2}

Subtract

(5 sin (X))^{2}

from both sides

(1 - cos (X))^{2} - 25 sin^{2} (X) = 0

Rewrite using trig identities

(1 - cos (X))^{2} - 25 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} - 25 (1 - cos^{2} (X))

Simplify

(1 - cos (X))^{2} - 25 (1 - cos^{2} (X)) : 26 cos^{2} (X) - 2 cos (X) - 24

(1 - cos (X))^{2} - 25 (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) - 25 (1 - cos^{2} (X))

Expand

- 25 (1 - cos^{2} (X)) : - 25 + 25 cos^{2} (X)

- 25 (1 - cos^{2} (X))

Apply the distributive law:

a (b - c) = ab - a c

a = - 25, b = 1, c = cos^{2} (X)

= - 25 \cdot 1 - (- 25) cos^{2} (X)

Apply minus-plus rules

- (- a) = a

= - 25 \cdot 1 + 25 cos^{2} (X)

Multiply the numbers:

25 \cdot 1 = 25

= - 25 + 25 cos^{2} (X)

= 1 - 2 cos (X) + cos^{2} (X) - 25 + 25 cos^{2} (X)

Simplify

1 - 2 cos (X) + cos^{2} (X) - 25 + 25 cos^{2} (X) : 26 cos^{2} (X) - 2 cos (X) - 24

1 - 2 cos (X) + cos^{2} (X) - 25 + 25 cos^{2} (X)

Group like terms

= - 2 cos (X) + cos^{2} (X) + 25 cos^{2} (X) + 1 - 25

Add similar elements:

cos^{2} (X) + 25 cos^{2} (X) = 26 cos^{2} (X)

= - 2 cos (X) + 26 cos^{2} (X) + 1 - 25

Add/Subtract the numbers:

1 - 25 = - 24

= 26 cos^{2} (X) - 2 cos (X) - 24

= 26 cos^{2} (X) - 2 cos (X) - 24

= 26 cos^{2} (X) - 2 cos (X) - 24

- 24 + 26 cos^{2} (X) - 2 cos (X) = 0

Solve by substitution

- 24 + 26 cos^{2} (X) - 2 cos (X) = 0

Let:

cos (X) = u

- 24 + 26 u^{2} - 2 u = 0

- 24 + 26 u^{2} - 2 u = 0 : u = 1, u = - \frac{12}{13}

- 24 + 26 u^{2} - 2 u = 0

Write in the standard form

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

26 u^{2} - 2 u - 24 = 0

Solve with the quadratic formula

26 u^{2} - 2 u - 24 = 0

Quadratic Equation Formula:

For

a = 26, b = - 2, c = - 24

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 26 ( - 24 )}{2 \cdot 26}

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 26 ( - 24 )}{2 \cdot 26}

(- 2)^{2} - 4 \cdot 26 (- 24) = 50

(- 2)^{2} - 4 \cdot 26 (- 24)

Apply rule

- (- a) = a

= (- 2)^{2} + 4 \cdot 26 \cdot 24

Apply exponent rule:

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

n

is even

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

= 2^{2} + 4 \cdot 26 \cdot 24

Multiply the numbers:

4 \cdot 26 \cdot 24 = 2496

= 2^{2} + 2496

2^{2} = 4

= 4 + 2496

Add the numbers:

4 + 2496 = 2500

= 2500

Factor the number:

2500 = 5 0^{2}

= 5 0^{2}

Apply radical rule:

5 0^{2} = 50

= 50

u_{1, 2} = \frac{- ( - 2 ) \pm 50}{2 \cdot 26}

Separate the solutions

u_{1} = \frac{- ( - 2 ) + 50}{2 \cdot 26}, u_{2} = \frac{- ( - 2 ) - 50}{2 \cdot 26}

u = \frac{- ( - 2 ) + 50}{2 \cdot 26} : 1

\frac{- ( - 2 ) + 50}{2 \cdot 26}

Apply rule

- (- a) = a

= \frac{2 + 50}{2 \cdot 26}

Add the numbers:

2 + 50 = 52

= \frac{52}{2 \cdot 26}

Multiply the numbers:

2 \cdot 26 = 52

= \frac{52}{52}

Apply rule

\frac{a}{a} = 1

= 1

u = \frac{- ( - 2 ) - 50}{2 \cdot 26} : - \frac{12}{13}

\frac{- ( - 2 ) - 50}{2 \cdot 26}

Apply rule

- (- a) = a

= \frac{2 - 50}{2 \cdot 26}

Subtract the numbers:

2 - 50 = - 48

= \frac{- 48}{2 \cdot 26}

Multiply the numbers:

2 \cdot 26 = 52

= \frac{- 48}{52}

Apply the fraction rule:

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

= - \frac{48}{52}

Cancel the common factor:

4

= - \frac{12}{13}

The solutions to the quadratic equation are:

u = 1, u = - \frac{12}{13}

Substitute back

u = cos (X)

cos (X) = 1, cos (X) = - \frac{12}{13}

cos (X) = 1, cos (X) = - \frac{12}{13}

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{12}{13} : X = arccos (- \frac{12}{13}) + 2 πn, X = - arccos (- \frac{12}{13}) + 2 πn

cos (X) = - \frac{12}{13}

Apply trig inverse properties

cos (X) = - \frac{12}{13}

General solutions for

cos (X) = - \frac{12}{13}

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

X = arccos (- \frac{12}{13}) + 2 πn, X = - arccos (- \frac{12}{13}) + 2 πn

X = arccos (- \frac{12}{13}) + 2 πn, X = - arccos (- \frac{12}{13}) + 2 πn

Combine all the solutions

X = 2 πn, X = arccos (- \frac{12}{13}) + 2 πn, X = - arccos (- \frac{12}{13}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

csc (X) - cot (X) = 5

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) = 5

plug in

X = 2 π 1

csc (2 π 1) - cot (2 π 1) = 5

Undefined

\Rightarrow False

Check the solution

arccos (- \frac{12}{13}) + 2 πn :

True

arccos (- \frac{12}{13}) + 2 πn

Plug in

n = 1

arccos (- \frac{12}{13}) + 2 π 1

For

csc (X) - cot (X) = 5

plug in

X = arccos (- \frac{12}{13}) + 2 π 1

csc (arccos (- \frac{12}{13}) + 2 π 1) - cot (arccos (- \frac{12}{13}) + 2 π 1) = 5

Refine

5 = 5

\Rightarrow True

Check the solution

- arccos (- \frac{12}{13}) + 2 πn :

False

- arccos (- \frac{12}{13}) + 2 πn

Plug in

n = 1

- arccos (- \frac{12}{13}) + 2 π 1

For

csc (X) - cot (X) = 5

plug in

X = - arccos (- \frac{12}{13}) + 2 π 1

csc (- arccos (- \frac{12}{13}) + 2 π 1) - cot (- arccos (- \frac{12}{13}) + 2 π 1) = 5

Refine

- 5 = 5

\Rightarrow False

X = arccos (- \frac{12}{13}) + 2 πn

Show solutions in decimal form

X = 2.74680 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for csc(X)-cot(X)=5 ?
The general solution for csc(X)-cot(X)=5 is X=2.74680…+2pin