What is the general solution for cot(θ)+2csc(θ)=6 ?

The general solution for cot(θ)+2csc(θ)=6 is θ=0.50017…+2pin,θ=2.97171…+2pin

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Popular Trigonometry >

cot(θ)+2csc(θ)=6

Solution

cot (θ) + 2 csc (θ) = 6

Solution

θ = 0.50017 \dots + 2 πn, θ = 2.97171 \dots + 2 πn

Degrees

θ = 28.65815 \dots^{\circ} + 36 0^{\circ} n, θ = 170.26648 \dots^{\circ} + 36 0^{\circ} n

Solution steps

cot (θ) + 2 csc (θ) = 6

Subtract

6

from both sides

cot (θ) + 2 csc (θ) - 6 = 0

Express with sin, cos

\frac{cos ( θ )}{sin ( θ )} + 2 \cdot \frac{1}{sin ( θ )} - 6 = 0

Simplify

\frac{cos ( θ )}{sin ( θ )} + 2 \cdot \frac{1}{sin ( θ )} - 6 : \frac{cos ( θ ) + 2 - 6 sin ( θ )}{sin ( θ )}

\frac{cos ( θ )}{sin ( θ )} + 2 \cdot \frac{1}{sin ( θ )} - 6

2 \cdot \frac{1}{sin ( θ )} = \frac{2}{sin ( θ )}

2 \cdot \frac{1}{sin ( θ )}

Multiply fractions:

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

= \frac{1 \cdot 2}{sin ( θ )}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{sin ( θ )}

= \frac{cos ( θ )}{sin ( θ )} + \frac{2}{sin ( θ )} - 6

Combine the fractions

\frac{cos ( θ )}{sin ( θ )} + \frac{2}{sin ( θ )} : \frac{cos ( θ ) + 2}{sin ( θ )}

Apply rule

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

= \frac{cos ( θ ) + 2}{sin ( θ )}

= \frac{cos ( θ ) + 2}{sin ( θ )} - 6

Convert element to fraction:

6 = \frac{6 sin ( θ )}{sin ( θ )}

= \frac{cos ( θ ) + 2}{sin ( θ )} - \frac{6 sin ( θ )}{sin ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{cos ( θ ) + 2 - 6 sin ( θ )}{sin ( θ )}

\frac{cos ( θ ) + 2 - 6 sin ( θ )}{sin ( θ )} = 0

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

cos (θ) + 2 - 6 sin (θ) = 0

Add

6 sin (θ)

to both sides

cos (θ) + 2 = 6 sin (θ)

Square both sides

(cos (θ) + 2)^{2} = (6 sin (θ))^{2}

Subtract

(6 sin (θ))^{2}

from both sides

(cos (θ) + 2)^{2} - 36 sin^{2} (θ) = 0

Rewrite using trig identities

(2 + cos (θ))^{2} - 36 sin^{2} (θ)

Use the Pythagorean identity:

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

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

= (2 + cos (θ))^{2} - 36 (1 - cos^{2} (θ))

Simplify

(2 + cos (θ))^{2} - 36 (1 - cos^{2} (θ)) : 37 cos^{2} (θ) + 4 cos (θ) - 32

(2 + cos (θ))^{2} - 36 (1 - cos^{2} (θ))

(2 + cos (θ))^{2} : 4 + 4 cos (θ) + cos^{2} (θ)

Apply Perfect Square Formula:

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

a = 2, b = cos (θ)

= 2^{2} + 2 \cdot 2 cos (θ) + cos^{2} (θ)

Simplify

2^{2} + 2 \cdot 2 cos (θ) + cos^{2} (θ) : 4 + 4 cos (θ) + cos^{2} (θ)

2^{2} + 2 \cdot 2 cos (θ) + cos^{2} (θ)

2^{2} = 4

= 4 + 2 \cdot 2 cos (θ) + cos^{2} (θ)

Multiply the numbers:

2 \cdot 2 = 4

= 4 + 4 cos (θ) + cos^{2} (θ)

= 4 + 4 cos (θ) + cos^{2} (θ)

= 4 + 4 cos (θ) + cos^{2} (θ) - 36 (1 - cos^{2} (θ))

Expand

- 36 (1 - cos^{2} (θ)) : - 36 + 36 cos^{2} (θ)

- 36 (1 - cos^{2} (θ))

Apply the distributive law:

a (b - c) = ab - a c

a = - 36, b = 1, c = cos^{2} (θ)

= - 36 \cdot 1 - (- 36) cos^{2} (θ)

Apply minus-plus rules

- (- a) = a

= - 36 \cdot 1 + 36 cos^{2} (θ)

Multiply the numbers:

36 \cdot 1 = 36

= - 36 + 36 cos^{2} (θ)

= 4 + 4 cos (θ) + cos^{2} (θ) - 36 + 36 cos^{2} (θ)

Simplify

4 + 4 cos (θ) + cos^{2} (θ) - 36 + 36 cos^{2} (θ) : 37 cos^{2} (θ) + 4 cos (θ) - 32

4 + 4 cos (θ) + cos^{2} (θ) - 36 + 36 cos^{2} (θ)

Group like terms

= 4 cos (θ) + cos^{2} (θ) + 36 cos^{2} (θ) + 4 - 36

Add similar elements:

cos^{2} (θ) + 36 cos^{2} (θ) = 37 cos^{2} (θ)

= 4 cos (θ) + 37 cos^{2} (θ) + 4 - 36

Add/Subtract the numbers:

4 - 36 = - 32

= 37 cos^{2} (θ) + 4 cos (θ) - 32

= 37 cos^{2} (θ) + 4 cos (θ) - 32

= 37 cos^{2} (θ) + 4 cos (θ) - 32

- 32 + 37 cos^{2} (θ) + 4 cos (θ) = 0

Solve by substitution

- 32 + 37 cos^{2} (θ) + 4 cos (θ) = 0

Let:

cos (θ) = u

- 32 + 37 u^{2} + 4 u = 0

- 32 + 37 u^{2} + 4 u = 0 : u = \frac{2 ( 3 33 - 1 )}{37}, u = - \frac{2 ( 1 + 3 33 )}{37}

- 32 + 37 u^{2} + 4 u = 0

Write in the standard form

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

37 u^{2} + 4 u - 32 = 0

Solve with the quadratic formula

37 u^{2} + 4 u - 32 = 0

Quadratic Equation Formula:

For

a = 37, b = 4, c = - 32

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 37 ( - 32 )}{2 \cdot 37}

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 37 ( - 32 )}{2 \cdot 37}

4^{2} - 4 \cdot 37 (- 32) = 1233

4^{2} - 4 \cdot 37 (- 32)

Apply rule

- (- a) = a

= 4^{2} + 4 \cdot 37 \cdot 32

Multiply the numbers:

4 \cdot 37 \cdot 32 = 4736

= 4^{2} + 4736

4^{2} = 16

= 16 + 4736

Add the numbers:

16 + 4736 = 4752

= 4752

Prime factorization of

4752 : 2^{4} \cdot 3^{3} \cdot 11

4752

4752

divides by

24752 = 2376 \cdot 2

= 2 \cdot 2376

2376

divides by

22376 = 1188 \cdot 2

= 2 \cdot 2 \cdot 1188

1188

divides by

21188 = 594 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 594

594

divides by

2594 = 297 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 297

297

divides by

3297 = 99 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 99

99

divides by

399 = 33 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 33

33

divides by

333 = 11 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 11

2, 3, 11

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 11

= 2^{4} \cdot 3^{3} \cdot 11

= 2^{4} \cdot 3^{3} \cdot 11

Apply exponent rule:

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

= 2^{4} \cdot 3^{2} \cdot 3 \cdot 11

Apply radical rule:

= 2^{4} 3^{2} 3 \cdot 11

Apply radical rule:

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 3^{2} 3 \cdot 11

Apply radical rule:

3^{2} = 3

= 2^{2} \cdot 3 3 \cdot 11

Refine

= 1233

u_{1, 2} = \frac{- 4 \pm 12 33}{2 \cdot 37}

Separate the solutions

u_{1} = \frac{- 4 + 12 33}{2 \cdot 37}, u_{2} = \frac{- 4 - 12 33}{2 \cdot 37}

u = \frac{- 4 + 12 33}{2 \cdot 37} : \frac{2 ( 3 33 - 1 )}{37}

\frac{- 4 + 12 33}{2 \cdot 37}

Multiply the numbers:

2 \cdot 37 = 74

= \frac{- 4 + 12 33}{74}

Factor

- 4 + 1233 : 4 (- 1 + 333)

- 4 + 1233

Rewrite as

= - 4 \cdot 1 + 4 \cdot 333

Factor out common term

4

= 4 (- 1 + 333)

= \frac{4 ( - 1 + 3 33 )}{74}

Cancel the common factor:

2

= \frac{2 ( 3 33 - 1 )}{37}

u = \frac{- 4 - 12 33}{2 \cdot 37} : - \frac{2 ( 1 + 3 33 )}{37}

\frac{- 4 - 12 33}{2 \cdot 37}

Multiply the numbers:

2 \cdot 37 = 74

= \frac{- 4 - 12 33}{74}

Factor

- 4 - 1233 : - 4 (1 + 333)

- 4 - 1233

Rewrite as

= - 4 \cdot 1 - 4 \cdot 333

Factor out common term

4

= - 4 (1 + 333)

= - \frac{4 ( 1 + 3 33 )}{74}

Cancel the common factor:

2

= - \frac{2 ( 1 + 3 33 )}{37}

The solutions to the quadratic equation are:

u = \frac{2 ( 3 33 - 1 )}{37}, u = - \frac{2 ( 1 + 3 33 )}{37}

Substitute back

u = cos (θ)

cos (θ) = \frac{2 ( 3 33 - 1 )}{37}, cos (θ) = - \frac{2 ( 1 + 3 33 )}{37}

cos (θ) = \frac{2 ( 3 33 - 1 )}{37}, cos (θ) = - \frac{2 ( 1 + 3 33 )}{37}

cos (θ) = \frac{2 ( 3 33 - 1 )}{37} : θ = arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 πn, θ = 2 π - arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 πn

cos (θ) = \frac{2 ( 3 33 - 1 )}{37}

Apply trig inverse properties

cos (θ) = \frac{2 ( 3 33 - 1 )}{37}

General solutions for

cos (θ) = \frac{2 ( 3 33 - 1 )}{37}

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

θ = arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 πn, θ = 2 π - arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 πn

θ = arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 πn, θ = 2 π - arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 πn

cos (θ) = - \frac{2 ( 1 + 3 33 )}{37} : θ = arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 πn, θ = - arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 πn

cos (θ) = - \frac{2 ( 1 + 3 33 )}{37}

Apply trig inverse properties

cos (θ) = - \frac{2 ( 1 + 3 33 )}{37}

General solutions for

cos (θ) = - \frac{2 ( 1 + 3 33 )}{37}

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

θ = arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 πn, θ = - arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 πn

θ = arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 πn, θ = - arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 πn

Combine all the solutions

θ = arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 πn, θ = 2 π - arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 πn, θ = arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 πn, θ = - arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

cot (θ) + 2 csc (θ) = 6

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

Check the solution

arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 πn :

True

arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 πn

Plug in

n = 1

arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 π 1

For

cot (θ) + 2 csc (θ) = 6

plug in

θ = arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 π 1

cot (arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 π 1) + 2 csc (arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 π 1) = 6

Refine

6 = 6

\Rightarrow True

Check the solution

2 π - arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 πn :

False

2 π - arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 π 1

For

cot (θ) + 2 csc (θ) = 6

plug in

θ = 2 π - arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 π 1

cot (2 π - arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 π 1) + 2 csc (2 π - arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 π 1) = 6

Refine

- 6 = 6

\Rightarrow False

Check the solution

arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 πn :

True

arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 πn

Plug in

n = 1

arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 π 1

For

cot (θ) + 2 csc (θ) = 6

plug in

θ = arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 π 1

cot (arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 π 1) + 2 csc (arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 π 1) = 6

Refine

6 = 6

\Rightarrow True

Check the solution

- arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 πn :

False

- arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 πn

Plug in

n = 1

- arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 π 1

For

cot (θ) + 2 csc (θ) = 6

plug in

θ = - arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 π 1

cot (- arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 π 1) + 2 csc (- arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 π 1) = 6

Refine

- 6 = 6

\Rightarrow False

θ = arccos (\frac{2 ( 3 33 - 1 )}{37}) + 2 πn, θ = arccos (- \frac{2 ( 1 + 3 33 )}{37}) + 2 πn

Show solutions in decimal form

θ = 0.50017 \dots + 2 πn, θ = 2.97171 \dots + 2 πn

Graph

Popular Examples

(1+cot(x))/(1+tan(x))=5

Frequently Asked Questions (FAQ)

What is the general solution for cot(θ)+2csc(θ)=6 ?
The general solution for cot(θ)+2csc(θ)=6 is θ=0.50017…+2pin,θ=2.97171…+2pin