What is the general solution for cot(θ)+3csc(θ)=5 ?

The general solution for cot(θ)+3csc(θ)=5 is θ=0.82641…+2pin,θ=2.70997…+2pin

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Popular Trigonometry >

cot(θ)+3csc(θ)=5

Solution

cot (θ) + 3 csc (θ) = 5

Solution

θ = 0.82641 \dots + 2 πn, θ = 2.70997 \dots + 2 πn

Degrees

θ = 47.34982 \dots^{\circ} + 36 0^{\circ} n, θ = 155.27003 \dots^{\circ} + 36 0^{\circ} n

Solution steps

cot (θ) + 3 csc (θ) = 5

Subtract

5

from both sides

cot (θ) + 3 csc (θ) - 5 = 0

Express with sin, cos

\frac{cos ( θ )}{sin ( θ )} + 3 \cdot \frac{1}{sin ( θ )} - 5 = 0

Simplify

\frac{cos ( θ )}{sin ( θ )} + 3 \cdot \frac{1}{sin ( θ )} - 5 : \frac{cos ( θ ) + 3 - 5 sin ( θ )}{sin ( θ )}

\frac{cos ( θ )}{sin ( θ )} + 3 \cdot \frac{1}{sin ( θ )} - 5

3 \cdot \frac{1}{sin ( θ )} = \frac{3}{sin ( θ )}

3 \cdot \frac{1}{sin ( θ )}

Multiply fractions:

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

= \frac{1 \cdot 3}{sin ( θ )}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{sin ( θ )}

= \frac{cos ( θ )}{sin ( θ )} + \frac{3}{sin ( θ )} - 5

Combine the fractions

\frac{cos ( θ )}{sin ( θ )} + \frac{3}{sin ( θ )} : \frac{cos ( θ ) + 3}{sin ( θ )}

Apply rule

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

= \frac{cos ( θ ) + 3}{sin ( θ )}

= \frac{cos ( θ ) + 3}{sin ( θ )} - 5

Convert element to fraction:

5 = \frac{5 sin ( θ )}{sin ( θ )}

= \frac{cos ( θ ) + 3}{sin ( θ )} - \frac{5 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 ) + 3 - 5 sin ( θ )}{sin ( θ )}

\frac{cos ( θ ) + 3 - 5 sin ( θ )}{sin ( θ )} = 0

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

cos (θ) + 3 - 5 sin (θ) = 0

Add

5 sin (θ)

to both sides

cos (θ) + 3 = 5 sin (θ)

Square both sides

(cos (θ) + 3)^{2} = (5 sin (θ))^{2}

Subtract

(5 sin (θ))^{2}

from both sides

(cos (θ) + 3)^{2} - 25 sin^{2} (θ) = 0

Rewrite using trig identities

(3 + cos (θ))^{2} - 25 sin^{2} (θ)

Use the Pythagorean identity:

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

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

= (3 + cos (θ))^{2} - 25 (1 - cos^{2} (θ))

Simplify

(3 + cos (θ))^{2} - 25 (1 - cos^{2} (θ)) : 26 cos^{2} (θ) + 6 cos (θ) - 16

(3 + cos (θ))^{2} - 25 (1 - cos^{2} (θ))

(3 + cos (θ))^{2} : 9 + 6 cos (θ) + cos^{2} (θ)

Apply Perfect Square Formula:

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

a = 3, b = cos (θ)

= 3^{2} + 2 \cdot 3 cos (θ) + cos^{2} (θ)

Simplify

3^{2} + 2 \cdot 3 cos (θ) + cos^{2} (θ) : 9 + 6 cos (θ) + cos^{2} (θ)

3^{2} + 2 \cdot 3 cos (θ) + cos^{2} (θ)

3^{2} = 9

= 9 + 2 \cdot 3 cos (θ) + cos^{2} (θ)

Multiply the numbers:

2 \cdot 3 = 6

= 9 + 6 cos (θ) + cos^{2} (θ)

= 9 + 6 cos (θ) + cos^{2} (θ)

= 9 + 6 cos (θ) + cos^{2} (θ) - 25 (1 - cos^{2} (θ))

Expand

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

- 25 (1 - cos^{2} (θ))

Apply the distributive law:

a (b - c) = ab - a c

a = - 25, b = 1, c = cos^{2} (θ)

= - 25 \cdot 1 - (- 25) cos^{2} (θ)

Apply minus-plus rules

- (- a) = a

= - 25 \cdot 1 + 25 cos^{2} (θ)

Multiply the numbers:

25 \cdot 1 = 25

= - 25 + 25 cos^{2} (θ)

= 9 + 6 cos (θ) + cos^{2} (θ) - 25 + 25 cos^{2} (θ)

Simplify

9 + 6 cos (θ) + cos^{2} (θ) - 25 + 25 cos^{2} (θ) : 26 cos^{2} (θ) + 6 cos (θ) - 16

9 + 6 cos (θ) + cos^{2} (θ) - 25 + 25 cos^{2} (θ)

Group like terms

= 6 cos (θ) + cos^{2} (θ) + 25 cos^{2} (θ) + 9 - 25

Add similar elements:

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

= 6 cos (θ) + 26 cos^{2} (θ) + 9 - 25

Add/Subtract the numbers:

9 - 25 = - 16

= 26 cos^{2} (θ) + 6 cos (θ) - 16

= 26 cos^{2} (θ) + 6 cos (θ) - 16

= 26 cos^{2} (θ) + 6 cos (θ) - 16

- 16 + 26 cos^{2} (θ) + 6 cos (θ) = 0

Solve by substitution

- 16 + 26 cos^{2} (θ) + 6 cos (θ) = 0

Let:

cos (θ) = u

- 16 + 26 u^{2} + 6 u = 0

- 16 + 26 u^{2} + 6 u = 0 : u = \frac{- 3 + 5 17}{26}, u = - \frac{3 + 5 17}{26}

- 16 + 26 u^{2} + 6 u = 0

Write in the standard form

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

26 u^{2} + 6 u - 16 = 0

Solve with the quadratic formula

26 u^{2} + 6 u - 16 = 0

Quadratic Equation Formula:

For

a = 26, b = 6, c = - 16

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 26 ( - 16 )}{2 \cdot 26}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 26 ( - 16 )}{2 \cdot 26}

6^{2} - 4 \cdot 26 (- 16) = 1017

6^{2} - 4 \cdot 26 (- 16)

Apply rule

- (- a) = a

= 6^{2} + 4 \cdot 26 \cdot 16

Multiply the numbers:

4 \cdot 26 \cdot 16 = 1664

= 6^{2} + 1664

6^{2} = 36

= 36 + 1664

Add the numbers:

36 + 1664 = 1700

= 1700

Prime factorization of

1700 : 2^{2} \cdot 5^{2} \cdot 17

1700

1700

divides by

21700 = 850 \cdot 2

= 2 \cdot 850

850

divides by

2850 = 425 \cdot 2

= 2 \cdot 2 \cdot 425

425

divides by

5425 = 85 \cdot 5

= 2 \cdot 2 \cdot 5 \cdot 85

85

divides by

585 = 17 \cdot 5

= 2 \cdot 2 \cdot 5 \cdot 5 \cdot 17

2, 5, 17

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 5 \cdot 5 \cdot 17

= 2^{2} \cdot 5^{2} \cdot 17

= 2^{2} \cdot 5^{2} \cdot 17

Apply radical rule:

= 17 2^{2} 5^{2}

Apply radical rule:

2^{2} = 2

= 217 5^{2}

Apply radical rule:

5^{2} = 5

= 2 \cdot 517

Refine

= 1017

u_{1, 2} = \frac{- 6 \pm 10 17}{2 \cdot 26}

Separate the solutions

u_{1} = \frac{- 6 + 10 17}{2 \cdot 26}, u_{2} = \frac{- 6 - 10 17}{2 \cdot 26}

u = \frac{- 6 + 10 17}{2 \cdot 26} : \frac{- 3 + 5 17}{26}

\frac{- 6 + 10 17}{2 \cdot 26}

Multiply the numbers:

2 \cdot 26 = 52

= \frac{- 6 + 10 17}{52}

Factor

- 6 + 1017 : 2 (- 3 + 517)

- 6 + 1017

Rewrite as

= - 2 \cdot 3 + 2 \cdot 517

Factor out common term

2

= 2 (- 3 + 517)

= \frac{2 ( - 3 + 5 17 )}{52}

Cancel the common factor:

2

= \frac{- 3 + 5 17}{26}

u = \frac{- 6 - 10 17}{2 \cdot 26} : - \frac{3 + 5 17}{26}

\frac{- 6 - 10 17}{2 \cdot 26}

Multiply the numbers:

2 \cdot 26 = 52

= \frac{- 6 - 10 17}{52}

Factor

- 6 - 1017 : - 2 (3 + 517)

- 6 - 1017

Rewrite as

= - 2 \cdot 3 - 2 \cdot 517

Factor out common term

2

= - 2 (3 + 517)

= - \frac{2 ( 3 + 5 17 )}{52}

Cancel the common factor:

2

= - \frac{3 + 5 17}{26}

The solutions to the quadratic equation are:

u = \frac{- 3 + 5 17}{26}, u = - \frac{3 + 5 17}{26}

Substitute back

u = cos (θ)

cos (θ) = \frac{- 3 + 5 17}{26}, cos (θ) = - \frac{3 + 5 17}{26}

cos (θ) = \frac{- 3 + 5 17}{26}, cos (θ) = - \frac{3 + 5 17}{26}

cos (θ) = \frac{- 3 + 5 17}{26} : θ = arccos (\frac{- 3 + 5 17}{26}) + 2 πn, θ = 2 π - arccos (\frac{- 3 + 5 17}{26}) + 2 πn

cos (θ) = \frac{- 3 + 5 17}{26}

Apply trig inverse properties

cos (θ) = \frac{- 3 + 5 17}{26}

General solutions for

cos (θ) = \frac{- 3 + 5 17}{26}

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

θ = arccos (\frac{- 3 + 5 17}{26}) + 2 πn, θ = 2 π - arccos (\frac{- 3 + 5 17}{26}) + 2 πn

θ = arccos (\frac{- 3 + 5 17}{26}) + 2 πn, θ = 2 π - arccos (\frac{- 3 + 5 17}{26}) + 2 πn

cos (θ) = - \frac{3 + 5 17}{26} : θ = arccos (- \frac{3 + 5 17}{26}) + 2 πn, θ = - arccos (- \frac{3 + 5 17}{26}) + 2 πn

cos (θ) = - \frac{3 + 5 17}{26}

Apply trig inverse properties

cos (θ) = - \frac{3 + 5 17}{26}

General solutions for

cos (θ) = - \frac{3 + 5 17}{26}

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

θ = arccos (- \frac{3 + 5 17}{26}) + 2 πn, θ = - arccos (- \frac{3 + 5 17}{26}) + 2 πn

θ = arccos (- \frac{3 + 5 17}{26}) + 2 πn, θ = - arccos (- \frac{3 + 5 17}{26}) + 2 πn

Combine all the solutions

θ = arccos (\frac{- 3 + 5 17}{26}) + 2 πn, θ = 2 π - arccos (\frac{- 3 + 5 17}{26}) + 2 πn, θ = arccos (- \frac{3 + 5 17}{26}) + 2 πn, θ = - arccos (- \frac{3 + 5 17}{26}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

cot (θ) + 3 csc (θ) = 5

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

Check the solution

arccos (\frac{- 3 + 5 17}{26}) + 2 πn :

True

arccos (\frac{- 3 + 5 17}{26}) + 2 πn

Plug in

n = 1

arccos (\frac{- 3 + 5 17}{26}) + 2 π 1

For

cot (θ) + 3 csc (θ) = 5

plug in

θ = arccos (\frac{- 3 + 5 17}{26}) + 2 π 1

cot (arccos (\frac{- 3 + 5 17}{26}) + 2 π 1) + 3 csc (arccos (\frac{- 3 + 5 17}{26}) + 2 π 1) = 5

Refine

5 = 5

\Rightarrow True

Check the solution

2 π - arccos (\frac{- 3 + 5 17}{26}) + 2 πn :

False

2 π - arccos (\frac{- 3 + 5 17}{26}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{- 3 + 5 17}{26}) + 2 π 1

For

cot (θ) + 3 csc (θ) = 5

plug in

θ = 2 π - arccos (\frac{- 3 + 5 17}{26}) + 2 π 1

cot (2 π - arccos (\frac{- 3 + 5 17}{26}) + 2 π 1) + 3 csc (2 π - arccos (\frac{- 3 + 5 17}{26}) + 2 π 1) = 5

Refine

- 5 = 5

\Rightarrow False

Check the solution

arccos (- \frac{3 + 5 17}{26}) + 2 πn :

True

arccos (- \frac{3 + 5 17}{26}) + 2 πn

Plug in

n = 1

arccos (- \frac{3 + 5 17}{26}) + 2 π 1

For

cot (θ) + 3 csc (θ) = 5

plug in

θ = arccos (- \frac{3 + 5 17}{26}) + 2 π 1

cot (arccos (- \frac{3 + 5 17}{26}) + 2 π 1) + 3 csc (arccos (- \frac{3 + 5 17}{26}) + 2 π 1) = 5

Refine

5 = 5

\Rightarrow True

Check the solution

- arccos (- \frac{3 + 5 17}{26}) + 2 πn :

False

- arccos (- \frac{3 + 5 17}{26}) + 2 πn

Plug in

n = 1

- arccos (- \frac{3 + 5 17}{26}) + 2 π 1

For

cot (θ) + 3 csc (θ) = 5

plug in

θ = - arccos (- \frac{3 + 5 17}{26}) + 2 π 1

cot (- arccos (- \frac{3 + 5 17}{26}) + 2 π 1) + 3 csc (- arccos (- \frac{3 + 5 17}{26}) + 2 π 1) = 5

Refine

- 5 = 5

\Rightarrow False

θ = arccos (\frac{- 3 + 5 17}{26}) + 2 πn, θ = arccos (- \frac{3 + 5 17}{26}) + 2 πn

Show solutions in decimal form

θ = 0.82641 \dots + 2 πn, θ = 2.70997 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cot(θ)+3csc(θ)=5 ?
The general solution for cot(θ)+3csc(θ)=5 is θ=0.82641…+2pin,θ=2.70997…+2pin