What is the general solution for 3cos(x)cot(x)+7=5csc(x) ?

The general solution for 3cos(x)cot(x)+7=5csc(x) is x=0.33983…+2pin,x=pi-0.33983…+2pin

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Popular Trigonometry >

3cos(x)cot(x)+7=5csc(x)

Solution

3 cos (x) cot (x) + 7 = 5 csc (x)

Solution

x = 0.33983 \dots + 2 πn, x = π - 0.33983 \dots + 2 πn

Degrees

x = 19.47122 \dots^{\circ} + 36 0^{\circ} n, x = 160.52877 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 cos (x) cot (x) + 7 = 5 csc (x)

Subtract

5 csc (x)

from both sides

3 cos (x) cot (x) + 7 - 5 csc (x) = 0

Express with sin, cos

3 cos (x) \frac{cos ( x )}{sin ( x )} + 7 - 5 \cdot \frac{1}{sin ( x )} = 0

Simplify

3 cos (x) \frac{cos ( x )}{sin ( x )} + 7 - 5 \cdot \frac{1}{sin ( x )} : \frac{3 cos ^{2} ( x ) - 5 + 7 sin ( x )}{sin ( x )}

3 cos (x) \frac{cos ( x )}{sin ( x )} + 7 - 5 \cdot \frac{1}{sin ( x )}

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

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

Multiply fractions:

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

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

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

cos (x) \cdot 3 cos (x)

Apply exponent rule:

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

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

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

Add the numbers:

1 + 1 = 2

= 3 cos^{2} (x)

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

5 \cdot \frac{1}{sin ( x )} = \frac{5}{sin ( x )}

5 \cdot \frac{1}{sin ( x )}

Multiply fractions:

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

= \frac{1 \cdot 5}{sin ( x )}

Multiply the numbers:

1 \cdot 5 = 5

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

= \frac{3 cos ^{2} ( x )}{sin ( x )} + 7 - \frac{5}{sin ( x )}

Combine the fractions

\frac{3 cos ^{2} ( x )}{sin ( x )} - \frac{5}{sin ( x )} : \frac{3 cos ^{2} ( x ) - 5}{sin ( x )}

Apply rule

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

= \frac{3 cos ^{2} ( x ) - 5}{sin ( x )}

= \frac{3 cos ^{2} ( x ) - 5}{sin ( x )} + 7

Convert element to fraction:

7 = \frac{7 sin ( x )}{sin ( x )}

= \frac{3 cos ^{2} ( x ) - 5}{sin ( x )} + \frac{7 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{3 cos ^{2} ( x ) - 5 + 7 sin ( x )}{sin ( x )}

\frac{3 cos ^{2} ( x ) - 5 + 7 sin ( x )}{sin ( x )} = 0

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

3 cos^{2} (x) - 5 + 7 sin (x) = 0

Subtract

7 sin (x)

from both sides

3 cos^{2} (x) - 5 = - 7 sin (x)

Square both sides

(3 cos^{2} (x) - 5)^{2} = (- 7 sin (x))^{2}

Subtract

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

from both sides

(3 cos^{2} (x) - 5)^{2} - 49 sin^{2} (x) = 0

Factor

(3 cos^{2} (x) - 5)^{2} - 49 sin^{2} (x) : (3 cos^{2} (x) - 5 + 7 sin (x)) (3 cos^{2} (x) - 5 - 7 sin (x))

(3 cos^{2} (x) - 5)^{2} - 49 sin^{2} (x)

Rewrite

(3 cos^{2} (x) - 5)^{2} - 49 sin^{2} (x)

(3 cos^{2} (x) - 5)^{2} - (7 sin (x))^{2}

(3 cos^{2} (x) - 5)^{2} - 49 sin^{2} (x)

Rewrite

49

7^{2}

= (3 cos^{2} (x) - 5)^{2} - 7^{2} sin^{2} (x)

Apply exponent rule:

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

7^{2} sin^{2} (x) = (7 sin (x))^{2}

= (3 cos^{2} (x) - 5)^{2} - (7 sin (x))^{2}

= (3 cos^{2} (x) - 5)^{2} - (7 sin (x))^{2}

Apply Difference of Two Squares Formula:

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

(3 cos^{2} (x) - 5)^{2} - (7 sin (x))^{2} = ((3 cos^{2} (x) - 5) + 7 sin (x)) ((3 cos^{2} (x) - 5) - 7 sin (x))

= ((3 cos^{2} (x) - 5) + 7 sin (x)) ((3 cos^{2} (x) - 5) - 7 sin (x))

Refine

= (3 cos^{2} (x) + 7 sin (x) - 5) (3 cos^{2} (x) - 7 sin (x) - 5)

(3 cos^{2} (x) - 5 + 7 sin (x)) (3 cos^{2} (x) - 5 - 7 sin (x)) = 0

Solving each part separately

3 cos^{2} (x) - 5 + 7 sin (x) = 0 or 3 cos^{2} (x) - 5 - 7 sin (x) = 0

3 cos^{2} (x) - 5 + 7 sin (x) = 0 : x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

3 cos^{2} (x) - 5 + 7 sin (x) = 0

Rewrite using trig identities

- 5 + 3 cos^{2} (x) + 7 sin (x)

Use the Pythagorean identity:

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

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

= - 5 + 3 (1 - sin^{2} (x)) + 7 sin (x)

Simplify

- 5 + 3 (1 - sin^{2} (x)) + 7 sin (x) : 7 sin (x) - 3 sin^{2} (x) - 2

- 5 + 3 (1 - sin^{2} (x)) + 7 sin (x)

Expand

3 (1 - sin^{2} (x)) : 3 - 3 sin^{2} (x)

3 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = 1, c = sin^{2} (x)

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

Multiply the numbers:

3 \cdot 1 = 3

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

= - 5 + 3 - 3 sin^{2} (x) + 7 sin (x)

Add/Subtract the numbers:

- 5 + 3 = - 2

= 7 sin (x) - 3 sin^{2} (x) - 2

= 7 sin (x) - 3 sin^{2} (x) - 2

- 2 - 3 sin^{2} (x) + 7 sin (x) = 0

Solve by substitution

- 2 - 3 sin^{2} (x) + 7 sin (x) = 0

Let:

sin (x) = u

- 2 - 3 u^{2} + 7 u = 0

- 2 - 3 u^{2} + 7 u = 0 : u = \frac{1}{3}, u = 2

- 2 - 3 u^{2} + 7 u = 0

Write in the standard form

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

- 3 u^{2} + 7 u - 2 = 0

Solve with the quadratic formula

- 3 u^{2} + 7 u - 2 = 0

Quadratic Equation Formula:

For

a = - 3, b = 7, c = - 2

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 ( - 3 ) ( - 2 )}{2 ( - 3 )}

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 ( - 3 ) ( - 2 )}{2 ( - 3 )}

7^{2} - 4 (- 3) (- 2) = 5

7^{2} - 4 (- 3) (- 2)

Apply rule

- (- a) = a

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

Multiply the numbers:

4 \cdot 3 \cdot 2 = 24

= 7^{2} - 24

7^{2} = 49

= 49 - 24

Subtract the numbers:

49 - 24 = 25

= 25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- 7 \pm 5}{2 ( - 3 )}

Separate the solutions

u_{1} = \frac{- 7 + 5}{2 ( - 3 )}, u_{2} = \frac{- 7 - 5}{2 ( - 3 )}

u = \frac{- 7 + 5}{2 ( - 3 )} : \frac{1}{3}

\frac{- 7 + 5}{2 ( - 3 )}

Remove parentheses:

(- a) = - a

= \frac{- 7 + 5}{- 2 \cdot 3}

Add/Subtract the numbers:

- 7 + 5 = - 2

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 2}{- 6}

Apply the fraction rule:

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

= \frac{2}{6}

Cancel the common factor:

2

= \frac{1}{3}

u = \frac{- 7 - 5}{2 ( - 3 )} : 2

\frac{- 7 - 5}{2 ( - 3 )}

Remove parentheses:

(- a) = - a

= \frac{- 7 - 5}{- 2 \cdot 3}

Subtract the numbers:

- 7 - 5 = - 12

= \frac{- 12}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 12}{- 6}

Apply the fraction rule:

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

= \frac{12}{6}

Divide the numbers:

\frac{12}{6} = 2

= 2

The solutions to the quadratic equation are:

u = \frac{1}{3}, u = 2

Substitute back

u = sin (x)

sin (x) = \frac{1}{3}, sin (x) = 2

sin (x) = \frac{1}{3}, sin (x) = 2

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

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

Apply trig inverse properties

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

General solutions for

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

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

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

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

sin (x) = 2 :

No Solution

sin (x) = 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

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

3 cos^{2} (x) - 5 - 7 sin (x) = 0 : x = arcsin (- \frac{1}{3}) + 2 πn, x = π + arcsin (\frac{1}{3}) + 2 πn

3 cos^{2} (x) - 5 - 7 sin (x) = 0

Rewrite using trig identities

- 5 + 3 cos^{2} (x) - 7 sin (x)

Use the Pythagorean identity:

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

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

= - 5 + 3 (1 - sin^{2} (x)) - 7 sin (x)

Simplify

- 5 + 3 (1 - sin^{2} (x)) - 7 sin (x) : - 3 sin^{2} (x) - 7 sin (x) - 2

- 5 + 3 (1 - sin^{2} (x)) - 7 sin (x)

Expand

3 (1 - sin^{2} (x)) : 3 - 3 sin^{2} (x)

3 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = 1, c = sin^{2} (x)

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

Multiply the numbers:

3 \cdot 1 = 3

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

= - 5 + 3 - 3 sin^{2} (x) - 7 sin (x)

Add/Subtract the numbers:

- 5 + 3 = - 2

= - 3 sin^{2} (x) - 7 sin (x) - 2

= - 3 sin^{2} (x) - 7 sin (x) - 2

- 2 - 3 sin^{2} (x) - 7 sin (x) = 0

Solve by substitution

- 2 - 3 sin^{2} (x) - 7 sin (x) = 0

Let:

sin (x) = u

- 2 - 3 u^{2} - 7 u = 0

- 2 - 3 u^{2} - 7 u = 0 : u = - 2, u = - \frac{1}{3}

- 2 - 3 u^{2} - 7 u = 0

Write in the standard form

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

- 3 u^{2} - 7 u - 2 = 0

Solve with the quadratic formula

- 3 u^{2} - 7 u - 2 = 0

Quadratic Equation Formula:

For

a = - 3, b = - 7, c = - 2

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 ( - 3 ) ( - 2 )}{2 ( - 3 )}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 ( - 3 ) ( - 2 )}{2 ( - 3 )}

(- 7)^{2} - 4 (- 3) (- 2) = 5

(- 7)^{2} - 4 (- 3) (- 2)

Apply rule

- (- a) = a

= (- 7)^{2} - 4 \cdot 3 \cdot 2

Apply exponent rule:

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

n

is even

(- 7)^{2} = 7^{2}

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

Multiply the numbers:

4 \cdot 3 \cdot 2 = 24

= 7^{2} - 24

7^{2} = 49

= 49 - 24

Subtract the numbers:

49 - 24 = 25

= 25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- ( - 7 ) \pm 5}{2 ( - 3 )}

Separate the solutions

u_{1} = \frac{- ( - 7 ) + 5}{2 ( - 3 )}, u_{2} = \frac{- ( - 7 ) - 5}{2 ( - 3 )}

u = \frac{- ( - 7 ) + 5}{2 ( - 3 )} : - 2

\frac{- ( - 7 ) + 5}{2 ( - 3 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{7 + 5}{- 2 \cdot 3}

Add the numbers:

7 + 5 = 12

= \frac{12}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{12}{- 6}

Apply the fraction rule:

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

= - \frac{12}{6}

Divide the numbers:

\frac{12}{6} = 2

= - 2

u = \frac{- ( - 7 ) - 5}{2 ( - 3 )} : - \frac{1}{3}

\frac{- ( - 7 ) - 5}{2 ( - 3 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{7 - 5}{- 2 \cdot 3}

Subtract the numbers:

7 - 5 = 2

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{2}{- 6}

Apply the fraction rule:

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

= - \frac{2}{6}

Cancel the common factor:

2

= - \frac{1}{3}

The solutions to the quadratic equation are:

u = - 2, u = - \frac{1}{3}

Substitute back

u = sin (x)

sin (x) = - 2, sin (x) = - \frac{1}{3}

sin (x) = - 2, sin (x) = - \frac{1}{3}

sin (x) = - 2 :

No Solution

sin (x) = - 2

- 1 \leq sin (x) \leq 1

No Solution

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

sin (x) = - \frac{1}{3}

Apply trig inverse properties

sin (x) = - \frac{1}{3}

General solutions for

sin (x) = - \frac{1}{3}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

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

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

Combine all the solutions

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

Combine all the solutions

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

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

3 cos (x) cot (x) + 7 = 5 csc (x)

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

Check the solution

arcsin (\frac{1}{3}) + 2 πn :

True

arcsin (\frac{1}{3}) + 2 πn

Plug in

n = 1

arcsin (\frac{1}{3}) + 2 π 1

For

3 cos (x) cot (x) + 7 = 5 csc (x)

plug in

x = arcsin (\frac{1}{3}) + 2 π 1

3 cos (arcsin (\frac{1}{3}) + 2 π 1) cot (arcsin (\frac{1}{3}) + 2 π 1) + 7 = 5 csc (arcsin (\frac{1}{3}) + 2 π 1)

Refine

15 = 15

\Rightarrow True

Check the solution

π - arcsin (\frac{1}{3}) + 2 πn :

True

π - arcsin (\frac{1}{3}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{1}{3}) + 2 π 1

For

3 cos (x) cot (x) + 7 = 5 csc (x)

plug in

x = π - arcsin (\frac{1}{3}) + 2 π 1

3 cos (π - arcsin (\frac{1}{3}) + 2 π 1) cot (π - arcsin (\frac{1}{3}) + 2 π 1) + 7 = 5 csc (π - arcsin (\frac{1}{3}) + 2 π 1)

Refine

15 = 15

\Rightarrow True

Check the solution

arcsin (- \frac{1}{3}) + 2 πn :

False

arcsin (- \frac{1}{3}) + 2 πn

Plug in

n = 1

arcsin (- \frac{1}{3}) + 2 π 1

For

3 cos (x) cot (x) + 7 = 5 csc (x)

plug in

x = arcsin (- \frac{1}{3}) + 2 π 1

3 cos (arcsin (- \frac{1}{3}) + 2 π 1) cot (arcsin (- \frac{1}{3}) + 2 π 1) + 7 = 5 csc (arcsin (- \frac{1}{3}) + 2 π 1)

Refine

- 1 = - 15

\Rightarrow False

Check the solution

π + arcsin (\frac{1}{3}) + 2 πn :

False

π + arcsin (\frac{1}{3}) + 2 πn

Plug in

n = 1

π + arcsin (\frac{1}{3}) + 2 π 1

For

3 cos (x) cot (x) + 7 = 5 csc (x)

plug in

x = π + arcsin (\frac{1}{3}) + 2 π 1

3 cos (π + arcsin (\frac{1}{3}) + 2 π 1) cot (π + arcsin (\frac{1}{3}) + 2 π 1) + 7 = 5 csc (π + arcsin (\frac{1}{3}) + 2 π 1)

Refine

- 1 = - 15

\Rightarrow False

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

Show solutions in decimal form

x = 0.33983 \dots + 2 πn, x = π - 0.33983 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 3cos(x)cot(x)+7=5csc(x) ?
The general solution for 3cos(x)cot(x)+7=5csc(x) is x=0.33983…+2pin,x=pi-0.33983…+2pin