What is the general solution for cot(θ)+4csc(θ)=6 ?

The general solution for cot(θ)+4csc(θ)=6 is θ=0.88277…+2pin,θ=2.58911…+2pin

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cot(θ)+4csc(θ)=6

Solution

cot (θ) + 4 csc (θ) = 6

Solution

θ = 0.88277 \dots + 2 πn, θ = 2.58911 \dots + 2 πn

Degrees

θ = 50.57910 \dots^{\circ} + 36 0^{\circ} n, θ = 148.34553 \dots^{\circ} + 36 0^{\circ} n

Solution steps

cot (θ) + 4 csc (θ) = 6

Subtract

6

from both sides

cot (θ) + 4 csc (θ) - 6 = 0

Express with sin, cos

\frac{cos ( θ )}{sin ( θ )} + 4 \cdot \frac{1}{sin ( θ )} - 6 = 0

Simplify

\frac{cos ( θ )}{sin ( θ )} + 4 \cdot \frac{1}{sin ( θ )} - 6 : \frac{cos ( θ ) + 4 - 6 sin ( θ )}{sin ( θ )}

\frac{cos ( θ )}{sin ( θ )} + 4 \cdot \frac{1}{sin ( θ )} - 6

4 \cdot \frac{1}{sin ( θ )} = \frac{4}{sin ( θ )}

4 \cdot \frac{1}{sin ( θ )}

Multiply fractions:

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

= \frac{1 \cdot 4}{sin ( θ )}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{sin ( θ )}

= \frac{cos ( θ )}{sin ( θ )} + \frac{4}{sin ( θ )} - 6

Combine the fractions

\frac{cos ( θ )}{sin ( θ )} + \frac{4}{sin ( θ )} : \frac{cos ( θ ) + 4}{sin ( θ )}

Apply rule

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

= \frac{cos ( θ ) + 4}{sin ( θ )}

= \frac{cos ( θ ) + 4}{sin ( θ )} - 6

Convert element to fraction:

6 = \frac{6 sin ( θ )}{sin ( θ )}

= \frac{cos ( θ ) + 4}{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 ) + 4 - 6 sin ( θ )}{sin ( θ )}

\frac{cos ( θ ) + 4 - 6 sin ( θ )}{sin ( θ )} = 0

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

cos (θ) + 4 - 6 sin (θ) = 0

Add

6 sin (θ)

to both sides

cos (θ) + 4 = 6 sin (θ)

Square both sides

(cos (θ) + 4)^{2} = (6 sin (θ))^{2}

Subtract

(6 sin (θ))^{2}

from both sides

(cos (θ) + 4)^{2} - 36 sin^{2} (θ) = 0

Rewrite using trig identities

(4 + 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)

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

Simplify

(4 + cos (θ))^{2} - 36 (1 - cos^{2} (θ)) : 37 cos^{2} (θ) + 8 cos (θ) - 20

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

(4 + cos (θ))^{2} : 16 + 8 cos (θ) + cos^{2} (θ)

Apply Perfect Square Formula:

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

a = 4, b = cos (θ)

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

Simplify

4^{2} + 2 \cdot 4 cos (θ) + cos^{2} (θ) : 16 + 8 cos (θ) + cos^{2} (θ)

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

4^{2} = 16

= 16 + 2 \cdot 4 cos (θ) + cos^{2} (θ)

Multiply the numbers:

2 \cdot 4 = 8

= 16 + 8 cos (θ) + cos^{2} (θ)

= 16 + 8 cos (θ) + cos^{2} (θ)

= 16 + 8 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)

= 16 + 8 cos (θ) + cos^{2} (θ) - 36 + 36 cos^{2} (θ)

Simplify

16 + 8 cos (θ) + cos^{2} (θ) - 36 + 36 cos^{2} (θ) : 37 cos^{2} (θ) + 8 cos (θ) - 20

16 + 8 cos (θ) + cos^{2} (θ) - 36 + 36 cos^{2} (θ)

Group like terms

= 8 cos (θ) + cos^{2} (θ) + 36 cos^{2} (θ) + 16 - 36

Add similar elements:

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

= 8 cos (θ) + 37 cos^{2} (θ) + 16 - 36

Add/Subtract the numbers:

16 - 36 = - 20

= 37 cos^{2} (θ) + 8 cos (θ) - 20

= 37 cos^{2} (θ) + 8 cos (θ) - 20

= 37 cos^{2} (θ) + 8 cos (θ) - 20

- 20 + 37 cos^{2} (θ) + 8 cos (θ) = 0

Solve by substitution

- 20 + 37 cos^{2} (θ) + 8 cos (θ) = 0

Let:

cos (θ) = u

- 20 + 37 u^{2} + 8 u = 0

- 20 + 37 u^{2} + 8 u = 0 : u = \frac{2 ( 3 21 - 2 )}{37}, u = - \frac{2 ( 2 + 3 21 )}{37}

- 20 + 37 u^{2} + 8 u = 0

Write in the standard form

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

37 u^{2} + 8 u - 20 = 0

Solve with the quadratic formula

37 u^{2} + 8 u - 20 = 0

Quadratic Equation Formula:

For

a = 37, b = 8, c = - 20

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 \cdot 37 ( - 20 )}{2 \cdot 37}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 \cdot 37 ( - 20 )}{2 \cdot 37}

8^{2} - 4 \cdot 37 (- 20) = 1221

8^{2} - 4 \cdot 37 (- 20)

Apply rule

- (- a) = a

= 8^{2} + 4 \cdot 37 \cdot 20

Multiply the numbers:

4 \cdot 37 \cdot 20 = 2960

= 8^{2} + 2960

8^{2} = 64

= 64 + 2960

Add the numbers:

64 + 2960 = 3024

= 3024

Prime factorization of

3024 : 2^{4} \cdot 3^{3} \cdot 7

3024

3024

divides by

23024 = 1512 \cdot 2

= 2 \cdot 1512

1512

divides by

21512 = 756 \cdot 2

= 2 \cdot 2 \cdot 756

756

divides by

2756 = 378 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 378

378

divides by

2378 = 189 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 189

189

divides by

3189 = 63 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 63

63

divides by

363 = 21 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 21

21

divides by

321 = 7 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 7

2, 3, 7

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 7

= 2^{4} \cdot 3^{3} \cdot 7

= 2^{4} \cdot 3^{3} \cdot 7

Apply exponent rule:

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

= 2^{4} \cdot 3^{2} \cdot 3 \cdot 7

Apply radical rule:

= 2^{4} 3^{2} 3 \cdot 7

Apply radical rule:

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

= 2^{2} 3^{2} 3 \cdot 7

Apply radical rule:

3^{2} = 3

= 2^{2} \cdot 3 3 \cdot 7

Refine

= 1221

u_{1, 2} = \frac{- 8 \pm 12 21}{2 \cdot 37}

Separate the solutions

u_{1} = \frac{- 8 + 12 21}{2 \cdot 37}, u_{2} = \frac{- 8 - 12 21}{2 \cdot 37}

u = \frac{- 8 + 12 21}{2 \cdot 37} : \frac{2 ( 3 21 - 2 )}{37}

\frac{- 8 + 12 21}{2 \cdot 37}

Multiply the numbers:

2 \cdot 37 = 74

= \frac{- 8 + 12 21}{74}

Factor

- 8 + 1221 : 4 (- 2 + 321)

- 8 + 1221

Rewrite as

= - 4 \cdot 2 + 4 \cdot 321

Factor out common term

4

= 4 (- 2 + 321)

= \frac{4 ( - 2 + 3 21 )}{74}

Cancel the common factor:

2

= \frac{2 ( 3 21 - 2 )}{37}

u = \frac{- 8 - 12 21}{2 \cdot 37} : - \frac{2 ( 2 + 3 21 )}{37}

\frac{- 8 - 12 21}{2 \cdot 37}

Multiply the numbers:

2 \cdot 37 = 74

= \frac{- 8 - 12 21}{74}

Factor

- 8 - 1221 : - 4 (2 + 321)

- 8 - 1221

Rewrite as

= - 4 \cdot 2 - 4 \cdot 321

Factor out common term

4

= - 4 (2 + 321)

= - \frac{4 ( 2 + 3 21 )}{74}

Cancel the common factor:

2

= - \frac{2 ( 2 + 3 21 )}{37}

The solutions to the quadratic equation are:

u = \frac{2 ( 3 21 - 2 )}{37}, u = - \frac{2 ( 2 + 3 21 )}{37}

Substitute back

u = cos (θ)

cos (θ) = \frac{2 ( 3 21 - 2 )}{37}, cos (θ) = - \frac{2 ( 2 + 3 21 )}{37}

cos (θ) = \frac{2 ( 3 21 - 2 )}{37}, cos (θ) = - \frac{2 ( 2 + 3 21 )}{37}

cos (θ) = \frac{2 ( 3 21 - 2 )}{37} : θ = arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 πn, θ = 2 π - arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 πn

cos (θ) = \frac{2 ( 3 21 - 2 )}{37}

Apply trig inverse properties

cos (θ) = \frac{2 ( 3 21 - 2 )}{37}

General solutions for

cos (θ) = \frac{2 ( 3 21 - 2 )}{37}

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

θ = arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 πn, θ = 2 π - arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 πn

θ = arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 πn, θ = 2 π - arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 πn

cos (θ) = - \frac{2 ( 2 + 3 21 )}{37} : θ = arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 πn, θ = - arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 πn

cos (θ) = - \frac{2 ( 2 + 3 21 )}{37}

Apply trig inverse properties

cos (θ) = - \frac{2 ( 2 + 3 21 )}{37}

General solutions for

cos (θ) = - \frac{2 ( 2 + 3 21 )}{37}

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

θ = arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 πn, θ = - arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 πn

θ = arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 πn, θ = - arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 πn

Combine all the solutions

θ = arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 πn, θ = 2 π - arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 πn, θ = arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 πn, θ = - arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

cot (θ) + 4 csc (θ) = 6

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

Check the solution

arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 πn :

True

arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 πn

Plug in

n = 1

arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 π 1

For

cot (θ) + 4 csc (θ) = 6

plug in

θ = arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 π 1

cot (arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 π 1) + 4 csc (arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 π 1) = 6

Refine

6 = 6

\Rightarrow True

Check the solution

2 π - arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 πn :

False

2 π - arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 π 1

For

cot (θ) + 4 csc (θ) = 6

plug in

θ = 2 π - arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 π 1

cot (2 π - arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 π 1) + 4 csc (2 π - arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 π 1) = 6

Refine

- 6 = 6

\Rightarrow False

Check the solution

arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 πn :

True

arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 πn

Plug in

n = 1

arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 π 1

For

cot (θ) + 4 csc (θ) = 6

plug in

θ = arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 π 1

cot (arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 π 1) + 4 csc (arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 π 1) = 6

Refine

6 = 6

\Rightarrow True

Check the solution

- arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 πn :

False

- arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 πn

Plug in

n = 1

- arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 π 1

For

cot (θ) + 4 csc (θ) = 6

plug in

θ = - arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 π 1

cot (- arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 π 1) + 4 csc (- arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 π 1) = 6

Refine

- 6 = 6

\Rightarrow False

θ = arccos (\frac{2 ( 3 21 - 2 )}{37}) + 2 πn, θ = arccos (- \frac{2 ( 2 + 3 21 )}{37}) + 2 πn

Show solutions in decimal form

θ = 0.88277 \dots + 2 πn, θ = 2.58911 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cot(θ)+4csc(θ)=6 ?
The general solution for cot(θ)+4csc(θ)=6 is θ=0.88277…+2pin,θ=2.58911…+2pin