What is the general solution for csc(x)cot(x)= 2/3 ?

The general solution for csc(x)cot(x)= 2/3 is x= pi/3+2pin,x=(5pi)/3+2pin

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csc(x)cot(x)= 2/3

Solution

csc (x) cot (x) = \frac{2}{3}

Solution

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

csc (x) cot (x) = \frac{2}{3}

Subtract

\frac{2}{3}

from both sides

csc (x) cot (x) - \frac{2}{3} = 0

Simplify

csc (x) cot (x) - \frac{2}{3} : \frac{3 csc ( x ) cot ( x ) - 2}{3}

csc (x) cot (x) - \frac{2}{3}

Convert element to fraction:

csc (x) cot (x) = \frac{csc ( x ) cot ( x ) 3}{3}

= \frac{csc ( x ) cot ( x ) \cdot 3}{3} - \frac{2}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{csc ( x ) cot ( x ) \cdot 3 - 2}{3}

\frac{3 csc ( x ) cot ( x ) - 2}{3} = 0

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

3 csc (x) cot (x) - 2 = 0

Express with sin, cos

3 \cdot \frac{1}{sin ( x )} \cdot \frac{cos ( x )}{sin ( x )} - 2 = 0

Simplify

3 \cdot \frac{1}{sin ( x )} \cdot \frac{cos ( x )}{sin ( x )} - 2 : \frac{3 cos ( x ) - 2 sin ^{2} ( x )}{sin ^{2} ( x )}

3 \cdot \frac{1}{sin ( x )} \cdot \frac{cos ( x )}{sin ( x )} - 2

3 \cdot \frac{1}{sin ( x )} \cdot \frac{cos ( x )}{sin ( x )} = \frac{3 cos ( x )}{sin ^{2} ( x )}

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 3 = 3

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

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

Apply exponent rule:

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

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

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

= \frac{3 cos ( x )}{sin ^{2} ( x )}

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

Convert element to fraction:

2 = \frac{2 sin ^{2} ( x )}{sin ^{2} ( x )}

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

Since the denominators are equal, combine the fractions:

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

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

\frac{3 cos ( x ) - 2 sin ^{2} ( x )}{sin ^{2} ( x )} = 0

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

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

Add

2 sin^{2} (x)

to both sides

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

Square both sides

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

Subtract

(2 sin^{2} (x))^{2}

from both sides

9 cos^{2} (x) - 4 sin^{4} (x) = 0

Factor

9 cos^{2} (x) - 4 sin^{4} (x) : (3 cos (x) + 2 sin^{2} (x)) (3 cos (x) - 2 sin^{2} (x))

9 cos^{2} (x) - 4 sin^{4} (x)

Rewrite

9 cos^{2} (x) - 4 sin^{4} (x)

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

9 cos^{2} (x) - 4 sin^{4} (x)

Rewrite

9

3^{2}

= 3^{2} cos^{2} (x) - 4 sin^{4} (x)

Rewrite

4

2^{2}

= 3^{2} cos^{2} (x) - 2^{2} sin^{4} (x)

Apply exponent rule:

a^{b c} = (a^{b})^{c}

sin^{4} (x) = (sin^{2} (x))^{2}

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

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

3^{2} cos^{2} (x) = (3 cos (x))^{2}

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

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

2^{2} (sin^{2} (x))^{2} = (2 sin^{2} (x))^{2}

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

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

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

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

Solving each part separately

3 cos (x) + 2 sin^{2} (x) = 0 or 3 cos (x) - 2 sin^{2} (x) = 0

3 cos (x) + 2 sin^{2} (x) = 0 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

3 cos (x) + 2 sin^{2} (x) = 0

Rewrite using trig identities

2 sin^{2} (x) + 3 cos (x)

Use the Pythagorean identity:

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

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

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

(1 - cos^{2} (x)) \cdot 2 + 3 cos (x) = 0

Solve by substitution

(1 - cos^{2} (x)) \cdot 2 + 3 cos (x) = 0

Let:

cos (x) = u

(1 - u^{2}) \cdot 2 + 3 u = 0

(1 - u^{2}) \cdot 2 + 3 u = 0 : u = - \frac{1}{2}, u = 2

(1 - u^{2}) \cdot 2 + 3 u = 0

Expand

(1 - u^{2}) \cdot 2 + 3 u : 2 - 2 u^{2} + 3 u

(1 - u^{2}) \cdot 2 + 3 u

= 2 (1 - u^{2}) + 3 u

Expand

2 (1 - u^{2}) : 2 - 2 u^{2}

2 (1 - u^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 1, c = u^{2}

= 2 \cdot 1 - 2 u^{2}

Multiply the numbers:

2 \cdot 1 = 2

= 2 - 2 u^{2}

= 2 - 2 u^{2} + 3 u

2 - 2 u^{2} + 3 u = 0

Write in the standard form

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

- 2 u^{2} + 3 u + 2 = 0

Solve with the quadratic formula

- 2 u^{2} + 3 u + 2 = 0

Quadratic Equation Formula:

For

a = - 2, b = 3, c = 2

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 2 ) \cdot 2}{2 ( - 2 )}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 2 ) \cdot 2}{2 ( - 2 )}

3^{2} - 4 (- 2) \cdot 2 = 5

3^{2} - 4 (- 2) \cdot 2

Apply rule

- (- a) = a

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

Multiply the numbers:

4 \cdot 2 \cdot 2 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

Add the numbers:

9 + 16 = 25

= 25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

5^{2} = 5

= 5

u_{1, 2} = \frac{- 3 \pm 5}{2 ( - 2 )}

Separate the solutions

u_{1} = \frac{- 3 + 5}{2 ( - 2 )}, u_{2} = \frac{- 3 - 5}{2 ( - 2 )}

u = \frac{- 3 + 5}{2 ( - 2 )} : - \frac{1}{2}

\frac{- 3 + 5}{2 ( - 2 )}

Remove parentheses:

(- a) = - a

= \frac{- 3 + 5}{- 2 \cdot 2}

Add/Subtract the numbers:

- 3 + 5 = 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{- 4}

Apply the fraction rule:

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

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

u = \frac{- 3 - 5}{2 ( - 2 )} : 2

\frac{- 3 - 5}{2 ( - 2 )}

Remove parentheses:

(- a) = - a

= \frac{- 3 - 5}{- 2 \cdot 2}

Subtract the numbers:

- 3 - 5 = - 8

= \frac{- 8}{- 2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 8}{- 4}

Apply the fraction rule:

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

= \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2

The solutions to the quadratic equation are:

u = - \frac{1}{2}, u = 2

Substitute back

u = cos (x)

cos (x) = - \frac{1}{2}, cos (x) = 2

cos (x) = - \frac{1}{2}, cos (x) = 2

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

cos (x) = - \frac{1}{2}

General solutions for

cos (x) = - \frac{1}{2}

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 = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

cos (x) = 2 :

No Solution

cos (x) = 2

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

3 cos (x) - 2 sin^{2} (x) = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

Rewrite using trig identities

- 2 sin^{2} (x) + 3 cos (x)

Use the Pythagorean identity:

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

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

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

- (1 - cos^{2} (x)) \cdot 2 + 3 cos (x) = 0

Solve by substitution

- (1 - cos^{2} (x)) \cdot 2 + 3 cos (x) = 0

Let:

cos (x) = u

- (1 - u^{2}) \cdot 2 + 3 u = 0

- (1 - u^{2}) \cdot 2 + 3 u = 0 : u = \frac{1}{2}, u = - 2

- (1 - u^{2}) \cdot 2 + 3 u = 0

Expand

- (1 - u^{2}) \cdot 2 + 3 u : - 2 + 2 u^{2} + 3 u

- (1 - u^{2}) \cdot 2 + 3 u

= - 2 (1 - u^{2}) + 3 u

Expand

- 2 (1 - u^{2}) : - 2 + 2 u^{2}

- 2 (1 - u^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = - 2, b = 1, c = u^{2}

= - 2 \cdot 1 - (- 2) u^{2}

Apply minus-plus rules

- (- a) = a

= - 2 \cdot 1 + 2 u^{2}

Multiply the numbers:

2 \cdot 1 = 2

= - 2 + 2 u^{2}

= - 2 + 2 u^{2} + 3 u

- 2 + 2 u^{2} + 3 u = 0

Write in the standard form

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

2 u^{2} + 3 u - 2 = 0

Solve with the quadratic formula

2 u^{2} + 3 u - 2 = 0

Quadratic Equation Formula:

For

a = 2, b = 3, c = - 2

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 2 ( - 2 )}{2 \cdot 2}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 2 ( - 2 )}{2 \cdot 2}

3^{2} - 4 \cdot 2 (- 2) = 5

3^{2} - 4 \cdot 2 (- 2)

Apply rule

- (- a) = a

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

Multiply the numbers:

4 \cdot 2 \cdot 2 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

Add the numbers:

9 + 16 = 25

= 25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

5^{2} = 5

= 5

u_{1, 2} = \frac{- 3 \pm 5}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- 3 + 5}{2 \cdot 2}, u_{2} = \frac{- 3 - 5}{2 \cdot 2}

u = \frac{- 3 + 5}{2 \cdot 2} : \frac{1}{2}

\frac{- 3 + 5}{2 \cdot 2}

Add/Subtract the numbers:

- 3 + 5 = 2

= \frac{2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

u = \frac{- 3 - 5}{2 \cdot 2} : - 2

\frac{- 3 - 5}{2 \cdot 2}

Subtract the numbers:

- 3 - 5 = - 8

= \frac{- 8}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 8}{4}

Apply the fraction rule:

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

= - \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= - 2

The solutions to the quadratic equation are:

u = \frac{1}{2}, u = - 2

Substitute back

u = cos (x)

cos (x) = \frac{1}{2}, cos (x) = - 2

cos (x) = \frac{1}{2}, cos (x) = - 2

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

cos (x) = \frac{1}{2}

General solutions for

cos (x) = \frac{1}{2}

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 = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

cos (x) = - 2 :

No Solution

cos (x) = - 2

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

csc (x) cot (x) = \frac{2}{3}

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

Check the solution

\frac{2 π}{3} + 2 πn :

False

\frac{2 π}{3} + 2 πn

Plug in

n = 1

\frac{2 π}{3} + 2 π 1

For

csc (x) cot (x) = \frac{2}{3}

plug in

x = \frac{2 π}{3} + 2 π 1

csc (\frac{2 π}{3} + 2 π 1) cot (\frac{2 π}{3} + 2 π 1) = \frac{2}{3}

Refine

- 0.66666 \dots = 0.66666 \dots

\Rightarrow False

Check the solution

\frac{4 π}{3} + 2 πn :

False

\frac{4 π}{3} + 2 πn

Plug in

n = 1

\frac{4 π}{3} + 2 π 1

For

csc (x) cot (x) = \frac{2}{3}

plug in

x = \frac{4 π}{3} + 2 π 1

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

Refine

- 0.66666 \dots = 0.66666 \dots

\Rightarrow False

Check the solution

\frac{π}{3} + 2 πn :

True

\frac{π}{3} + 2 πn

Plug in

n = 1

\frac{π}{3} + 2 π 1

For

csc (x) cot (x) = \frac{2}{3}

plug in

x = \frac{π}{3} + 2 π 1

csc (\frac{π}{3} + 2 π 1) cot (\frac{π}{3} + 2 π 1) = \frac{2}{3}

Refine

0.66666 \dots = 0.66666 \dots

\Rightarrow True

Check the solution

\frac{5 π}{3} + 2 πn :

True

\frac{5 π}{3} + 2 πn

Plug in

n = 1

\frac{5 π}{3} + 2 π 1

For

csc (x) cot (x) = \frac{2}{3}

plug in

x = \frac{5 π}{3} + 2 π 1

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

Refine

0.66666 \dots = 0.66666 \dots

\Rightarrow True

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for csc(x)cot(x)= 2/3 ?
The general solution for csc(x)cot(x)= 2/3 is x= pi/3+2pin,x=(5pi)/3+2pin