What is the general solution for 3tan(x)+cot(x)-5csc(x)=0 ?

The general solution for 3tan(x)+cot(x)-5csc(x)=0 is x= pi/3+2pin,x=(5pi)/3+2pin

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3tan(x)+cot(x)-5csc(x)=0

Solution

3 tan (x) + cot (x) - 5 csc (x) = 0

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

3 tan (x) + cot (x) - 5 csc (x) = 0

Express with sin, cos

cot (x) + 3 tan (x) - 5 csc (x)

Use the basic trigonometric identity:

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

= \frac{cos ( x )}{sin ( x )} + 3 tan (x) - 5 csc (x)

Use the basic trigonometric identity:

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

= \frac{cos ( x )}{sin ( x )} + 3 \cdot \frac{sin ( x )}{cos ( x )} - 5 csc (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Combine the fractions

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

Apply rule

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

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

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

Least Common Multiplier of

sin (x), cos (x) : sin (x) cos (x)

sin (x), cos (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (x)

cos (x)

= sin (x) cos (x)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

sin (x) cos (x)

For

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

multiply the denominator and numerator by

cos (x)

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

For

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

multiply the denominator and numerator by

sin (x)

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

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

Expand

cos (x) (- 5 + cos (x)) : - 5 cos (x) + cos^{2} (x)

cos (x) (- 5 + cos (x))

Apply the distributive law:

a (b + c) = ab + a c

a = cos (x), b = - 5, c = cos (x)

= cos (x) (- 5) + cos (x) cos (x)

Apply minus-plus rules

+ (- a) = - a

= - 5 cos (x) + cos (x) cos (x)

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

cos (x) cos (x)

Apply exponent rule:

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

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

= cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cos^{2} (x)

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Multiply the numbers:

3 \cdot 1 = 3

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

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

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

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

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

Solve by substitution

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

Let:

cos (x) = u

3 - 2 u^{2} - 5 u = 0

3 - 2 u^{2} - 5 u = 0 : u = - 3, u = \frac{1}{2}

3 - 2 u^{2} - 5 u = 0

Write in the standard form

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

- 2 u^{2} - 5 u + 3 = 0

Solve with the quadratic formula

- 2 u^{2} - 5 u + 3 = 0

Quadratic Equation Formula:

For

a = - 2, b = - 5, c = 3

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 2 ) \cdot 3}{2 ( - 2 )}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 2 ) \cdot 3}{2 ( - 2 )}

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

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 2 \cdot 3

Multiply the numbers:

4 \cdot 2 \cdot 3 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

Add the numbers:

25 + 24 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

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

Separate the solutions

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

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

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

Remove parentheses:

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

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

Add the numbers:

5 + 7 = 12

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{12}{- 4}

Apply the fraction rule:

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

= - \frac{12}{4}

Divide the numbers:

\frac{12}{4} = 3

= - 3

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

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

Remove parentheses:

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

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

Subtract the numbers:

5 - 7 = - 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}

The solutions to the quadratic equation are:

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

Substitute back

u = cos (x)

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

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

cos (x) = - 3 :

No Solution

cos (x) = - 3

- 1 \leq cos (x) \leq 1

No Solution

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

Combine all the solutions

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

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cos(c)=cos(7)cos(60)

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Frequently Asked Questions (FAQ)

What is the general solution for 3tan(x)+cot(x)-5csc(x)=0 ?
The general solution for 3tan(x)+cot(x)-5csc(x)=0 is x= pi/3+2pin,x=(5pi)/3+2pin