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3tan(x)+cot(x)<5sin(x)

Solution

3 tan (x) + cot (x) < 5 sin (x)

Solution

\frac{π}{2} + 2 πn < x < π + 2 πn or \frac{3 π}{2} + 2 πn < x < 2 π + 2 πn

Interval Notation

(\frac{π}{2} + 2 πn, π + 2 πn) \cup (\frac{3 π}{2} + 2 πn, 2 π + 2 πn)

Decimal Notation

1.57079 \dots + 2 πn < x < 3.14159 \dots + 2 πn or 4.71238 \dots + 2 πn < x < 6.28318 \dots + 2 πn

Solution steps

3 tan (x) + cot (x) < 5 sin (x)

Move

5 sin (x)

to the left side

3 tan (x) + cot (x) < 5 sin (x)

Subtract

5 sin (x)

from both sides

3 tan (x) + cot (x) - 5 sin (x) < 5 sin (x) - 5 sin (x)

3 tan (x) + cot (x) - 5 sin (x) < 0

3 tan (x) + cot (x) - 5 sin (x) < 0

Periodicity of

3 tan (x) + cot (x) - 5 sin (x) : 2 π

The compound periodicity of the sum of periodic functions is the least common multiplier of the periods

3 tan (x), cot (x), 5 sin (x)

Periodicity of

3 tan (x) : π

Periodicity of

a \cdot tan (b x + c) + d = \frac{periodicity of tan ( x )}{∣ b ∣}

Periodicity of

tan (x)

π

= \frac{π}{∣ 1 ∣}

Simplify

= π

Periodicity of

cot (x) : π

Periodicity of

cot (x)

π

= π

Periodicity of

5 sin (x) : 2 π

Periodicity of

a \cdot sin (b x + c) + d = \frac{periodicity of sin ( x )}{∣ b ∣}

Periodicity of

sin (x)

2 π

= \frac{2 π}{∣ 1 ∣}

Simplify

= 2 π

Combine periods:

π, π, 2 π

= 2 π

Express with sin, cos

3 tan (x) + cot (x) - 5 sin (x) < 0

Use the basic trigonometric identity:

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

3 \cdot \frac{sin ( x )}{cos ( x )} + cot (x) - 5 sin (x) < 0

Use the basic trigonometric identity:

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

3 \cdot \frac{sin ( x )}{cos ( x )} + \frac{cos ( x )}{sin ( x )} - 5 sin (x) < 0

3 \cdot \frac{sin ( x )}{cos ( x )} + \frac{cos ( x )}{sin ( x )} - 5 sin (x) < 0

Simplify

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

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

Multiply

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 )}

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

Convert element to fraction:

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

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

Least Common Multiplier of

cos (x), sin (x), 1 : cos (x) sin (x)

cos (x), sin (x), 1

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear in at least one of the factored expressions

= cos (x) sin (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

cos (x) sin (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 )}{cos ( x ) sin ( x )}

For

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

multiply the denominator and numerator by

cos (x)

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

For

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

multiply the denominator and numerator by

cos (x) sin (x)

\frac{5 sin ( x )}{1} = \frac{5 sin ( x ) cos ( x ) sin ( x )}{1 \cdot cos ( x ) sin ( x )} = \frac{5 sin ^{2} ( x ) cos ( x )}{cos ( x ) sin ( x )}

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

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

Find the zeroes and undifined points of

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

for

0 \leq x < 2 π

To find the zeroes, set the inequality to zero

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

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

No Solution for

x \in R

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = - 5 cos (x), b = 1, c = cos^{2} (x)

= - 5 cos (x) \cdot 1 - (- 5 cos (x)) cos^{2} (x)

Apply minus-plus rules

- (- a) = a

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

Simplify

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

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

5 \cdot 1 \cdot cos (x) = 5 cos (x)

5 \cdot 1 \cdot cos (x)

Multiply the numbers:

5 \cdot 1 = 5

= 5 cos (x)

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

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

Apply exponent rule:

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

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

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

Add the numbers:

2 + 1 = 3

= 5 cos^{3} (x)

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

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

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

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

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

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

Solve by substitution

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

Let:

cos (x) = u

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

3 - 2 u^{2} - 5 u + 5 u^{3} = 0 : u \approx - 1.06603 \dots

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

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

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

Find one solution for

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

using Newton-Raphson:

u \approx - 1.06603 \dots

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

Newton-Raphson Approximation Definition

f (u) = 5 u^{3} - 2 u^{2} - 5 u + 3

Find

f^{^{'}} (u) : 15 u^{2} - 4 u - 5

\frac{d}{d u} (5 u^{3} - 2 u^{2} - 5 u + 3)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (5 u^{3}) - \frac{d}{d u} (2 u^{2}) - \frac{d}{d u} (5 u) + \frac{d}{d u} (3)

\frac{d}{d u} (5 u^{3}) = 15 u^{2}

\frac{d}{d u} (5 u^{3})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 5 \frac{d}{d u} (u^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 5 \cdot 3 u^{3 - 1}

Simplify

= 15 u^{2}

\frac{d}{d u} (2 u^{2}) = 4 u

\frac{d}{d u} (2 u^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

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

Simplify

= 4 u

\frac{d}{d u} (5 u) = 5

\frac{d}{d u} (5 u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 5 \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 5 \cdot 1

Simplify

= 5

\frac{d}{d u} (3) = 0

\frac{d}{d u} (3)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 15 u^{2} - 4 u - 5 + 0

Simplify

= 15 u^{2} - 4 u - 5

Let

u_{0} = - 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 1.07142 \dots : Δ u_{1} = 0.07142 \dots

f (u_{0}) = 5 (- 1)^{3} - 2 (- 1)^{2} - 5 (- 1) + 3 = 1

f^{^{'}} (u_{0}) = 15 (- 1)^{2} - 4 (- 1) - 5 = 14

u_{1} = - 1.07142 \dots

Δ u_{1} = ∣ - 1.07142 \dots - (- 1) ∣ = 0.07142 \dots

Δ u_{1} = 0.07142 \dots

u_{2} = - 1.06606 \dots : Δ u_{2} = 0.00536 \dots

f (u_{1}) = 5 (- 1.07142 \dots)^{3} - 2 (- 1.07142 \dots)^{2} - 5 (- 1.07142 \dots) + 3 = - 0.08855 \dots

f^{^{'}} (u_{1}) = 15 (- 1.07142 \dots)^{2} - 4 (- 1.07142 \dots) - 5 = 16.50510 \dots

u_{2} = - 1.06606 \dots

Δ u_{2} = ∣ - 1.06606 \dots - (- 1.07142 \dots) ∣ = 0.00536 \dots

Δ u_{2} = 0.00536 \dots

u_{3} = - 1.06603 \dots : Δ u_{3} = 0.00003 \dots

f (u_{2}) = 5 (- 1.06606 \dots)^{3} - 2 (- 1.06606 \dots)^{2} - 5 (- 1.06606 \dots) + 3 = - 0.00051 \dots

f^{^{'}} (u_{2}) = 15 (- 1.06606 \dots)^{2} - 4 (- 1.06606 \dots) - 5 = 16.31161 \dots

u_{3} = - 1.06603 \dots

Δ u_{3} = ∣ - 1.06603 \dots - (- 1.06606 \dots) ∣ = 0.00003 \dots

Δ u_{3} = 0.00003 \dots

u_{4} = - 1.06603 \dots : Δ u_{4} = 1.11867 E - 9

f (u_{3}) = 5 (- 1.06603 \dots)^{3} - 2 (- 1.06603 \dots)^{2} - 5 (- 1.06603 \dots) + 3 = - 1.8246 E - 8

f^{^{'}} (u_{3}) = 15 (- 1.06603 \dots)^{2} - 4 (- 1.06603 \dots) - 5 = 16.31046 \dots

u_{4} = - 1.06603 \dots

Δ u_{4} = ∣ - 1.06603 \dots - (- 1.06603 \dots) ∣ = 1.11867 E - 9

Δ u_{4} = 1.11867 E - 9

u \approx - 1.06603 \dots

Apply long division:

\frac{5 u ^{3} - 2 u ^{2} - 5 u + 3}{u + 1.06603 \dots} = 5 u^{2} - 7.33015 \dots u + 2.81417 \dots

5 u^{2} - 7.33015 \dots u + 2.81417 \dots \approx 0

Find one solution for

5 u^{2} - 7.33015 \dots u + 2.81417 \dots = 0

using Newton-Raphson:

No Solution for

u \in R

5 u^{2} - 7.33015 \dots u + 2.81417 \dots = 0

Newton-Raphson Approximation Definition

f (u) = 5 u^{2} - 7.33015 \dots u + 2.81417 \dots

Find

f^{^{'}} (u) : 10 u - 7.33015 \dots

\frac{d}{d u} (5 u^{2} - 7.33015 \dots u + 2.81417 \dots)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (5 u^{2}) - \frac{d}{d u} (7.33015 \dots u) + \frac{d}{d u} (2.81417 \dots)

\frac{d}{d u} (5 u^{2}) = 10 u

\frac{d}{d u} (5 u^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 5 \frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 5 \cdot 2 u^{2 - 1}

Simplify

= 10 u

\frac{d}{d u} (7.33015 \dots u) = 7.33015 \dots

\frac{d}{d u} (7.33015 \dots u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 7.33015 \dots \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 7.33015 \dots \cdot 1

Simplify

= 7.33015 \dots

\frac{d}{d u} (2.81417 \dots) = 0

\frac{d}{d u} (2.81417 \dots)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 10 u - 7.33015 \dots + 0

Simplify

= 10 u - 7.33015 \dots

Let

u_{0} = 0

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 0.38391 \dots : Δ u_{1} = 0.38391 \dots

f (u_{0}) = 5 \cdot 0^{2} - 7.33015 \dots \cdot 0 + 2.81417 \dots = 2.81417 \dots

f^{^{'}} (u_{0}) = 10 \cdot 0 - 7.33015 \dots = - 7.33015 \dots

u_{1} = 0.38391 \dots

Δ u_{1} = ∣ 0.38391 \dots - 0 ∣ = 0.38391 \dots

Δ u_{1} = 0.38391 \dots

u_{2} = 0.59502 \dots : Δ u_{2} = 0.21110 \dots

f (u_{1}) = 5 \cdot 0.38391 \dots^{2} - 7.33015 \dots \cdot 0.38391 \dots + 2.81417 \dots = 0.73696 \dots

f^{^{'}} (u_{1}) = 10 \cdot 0.38391 \dots - 7.33015 \dots = - 3.49098 \dots

u_{2} = 0.59502 \dots

Δ u_{2} = ∣ 0.59502 \dots - 0.38391 \dots ∣ = 0.21110 \dots

Δ u_{2} = 0.21110 \dots

u_{3} = 0.75649 \dots : Δ u_{3} = 0.16147 \dots

f (u_{2}) = 5 \cdot 0.59502 \dots^{2} - 7.33015 \dots \cdot 0.59502 \dots + 2.81417 \dots = 0.22282 \dots

f^{^{'}} (u_{2}) = 10 \cdot 0.59502 \dots - 7.33015 \dots = - 1.37992 \dots

u_{3} = 0.75649 \dots

Δ u_{3} = ∣ 0.75649 \dots - 0.59502 \dots ∣ = 0.16147 \dots

Δ u_{3} = 0.16147 \dots

u_{4} = 0.20133 \dots : Δ u_{4} = 0.55516 \dots

f (u_{3}) = 5 \cdot 0.75649 \dots^{2} - 7.33015 \dots \cdot 0.75649 \dots + 2.81417 \dots = 0.13037 \dots

f^{^{'}} (u_{3}) = 10 \cdot 0.75649 \dots - 7.33015 \dots = 0.23484 \dots

u_{4} = 0.20133 \dots

Δ u_{4} = ∣ 0.20133 \dots - 0.75649 \dots ∣ = 0.55516 \dots

Δ u_{4} = 0.55516 \dots

u_{5} = 0.49118 \dots : Δ u_{5} = 0.28984 \dots

f (u_{4}) = 5 \cdot 0.20133 \dots^{2} - 7.33015 \dots \cdot 0.20133 \dots + 2.81417 \dots = 1.54101 \dots

f^{^{'}} (u_{4}) = 10 \cdot 0.20133 \dots - 7.33015 \dots = - 5.31676 \dots

u_{5} = 0.49118 \dots

Δ u_{5} = ∣ 0.49118 \dots - 0.20133 \dots ∣ = 0.28984 \dots

Δ u_{5} = 0.28984 \dots

u_{6} = 0.66486 \dots : Δ u_{6} = 0.17368 \dots

f (u_{5}) = 5 \cdot 0.49118 \dots^{2} - 7.33015 \dots \cdot 0.49118 \dots + 2.81417 \dots = 0.42003 \dots

f^{^{'}} (u_{5}) = 10 \cdot 0.49118 \dots - 7.33015 \dots = - 2.41835 \dots

u_{6} = 0.66486 \dots

Δ u_{6} = ∣ 0.66486 \dots - 0.49118 \dots ∣ = 0.17368 \dots

Δ u_{6} = 0.17368 \dots

u_{7} = 0.88620 \dots : Δ u_{7} = 0.22133 \dots

f (u_{6}) = 5 \cdot 0.66486 \dots^{2} - 7.33015 \dots \cdot 0.66486 \dots + 2.81417 \dots = 0.15083 \dots

f^{^{'}} (u_{6}) = 10 \cdot 0.66486 \dots - 7.33015 \dots = - 0.68147 \dots

u_{7} = 0.88620 \dots

Δ u_{7} = ∣ 0.88620 \dots - 0.66486 \dots ∣ = 0.22133 \dots

Δ u_{7} = 0.22133 \dots

u_{8} = 0.72630 \dots : Δ u_{8} = 0.15990 \dots

f (u_{7}) = 5 \cdot 0.88620 \dots^{2} - 7.33015 \dots \cdot 0.88620 \dots + 2.81417 \dots = 0.24495 \dots

f^{^{'}} (u_{7}) = 10 \cdot 0.88620 \dots - 7.33015 \dots = 1.53190 \dots

u_{8} = 0.72630 \dots

Δ u_{8} = ∣ 0.72630 \dots - 0.88620 \dots ∣ = 0.15990 \dots

Δ u_{8} = 0.15990 \dots

u_{9} = 2.63145 \dots : Δ u_{9} = 1.90514 \dots

f (u_{8}) = 5 \cdot 0.72630 \dots^{2} - 7.33015 \dots \cdot 0.72630 \dots + 2.81417 \dots = 0.12784 \dots

f^{^{'}} (u_{8}) = 10 \cdot 0.72630 \dots - 7.33015 \dots = - 0.06710 \dots

u_{9} = 2.63145 \dots

Δ u_{9} = ∣ 2.63145 \dots - 0.72630 \dots ∣ = 1.90514 \dots

Δ u_{9} = 1.90514 \dots

u_{10} = 1.67551 \dots : Δ u_{10} = 0.95594 \dots

f (u_{9}) = 5 \cdot 2.63145 \dots^{2} - 7.33015 \dots \cdot 2.63145 \dots + 2.81417 \dots = 18.14798 \dots

f^{^{'}} (u_{9}) = 10 \cdot 2.63145 \dots - 7.33015 \dots = 18.98439 \dots

u_{10} = 1.67551 \dots

Δ u_{10} = ∣ 1.67551 \dots - 2.63145 \dots ∣ = 0.95594 \dots

Δ u_{10} = 0.95594 \dots

u_{11} = 1.19072 \dots : Δ u_{11} = 0.48478 \dots

f (u_{10}) = 5 \cdot 1.67551 \dots^{2} - 7.33015 \dots \cdot 1.67551 \dots + 2.81417 \dots = 4.56912 \dots

f^{^{'}} (u_{10}) = 10 \cdot 1.67551 \dots - 7.33015 \dots = 9.42497 \dots

u_{11} = 1.19072 \dots

Δ u_{11} = ∣ 1.19072 \dots - 1.67551 \dots ∣ = 0.48478 \dots

Δ u_{11} = 0.48478 \dots

u_{12} = 0.93398 \dots : Δ u_{12} = 0.25673 \dots

f (u_{11}) = 5 \cdot 1.19072 \dots^{2} - 7.33015 \dots \cdot 1.19072 \dots + 2.81417 \dots = 1.17510 \dots

f^{^{'}} (u_{11}) = 10 \cdot 1.19072 \dots - 7.33015 \dots = 4.57708 \dots

u_{12} = 0.93398 \dots

Δ u_{12} = ∣ 0.93398 \dots - 1.19072 \dots ∣ = 0.25673 \dots

Δ u_{12} = 0.25673 \dots

u_{13} = 0.77000 \dots : Δ u_{13} = 0.16398 \dots

f (u_{12}) = 5 \cdot 0.93398 \dots^{2} - 7.33015 \dots \cdot 0.93398 \dots + 2.81417 \dots = 0.32956 \dots

f^{^{'}} (u_{12}) = 10 \cdot 0.93398 \dots - 7.33015 \dots = 2.00972 \dots

u_{13} = 0.77000 \dots

Δ u_{13} = ∣ 0.77000 \dots - 0.93398 \dots ∣ = 0.16398 \dots

Δ u_{13} = 0.16398 \dots

u_{14} = 0.40647 \dots : Δ u_{14} = 0.36352 \dots

f (u_{13}) = 5 \cdot 0.77000 \dots^{2} - 7.33015 \dots \cdot 0.77000 \dots + 2.81417 \dots = 0.13445 \dots

f^{^{'}} (u_{13}) = 10 \cdot 0.77000 \dots - 7.33015 \dots = 0.36986 \dots

u_{14} = 0.40647 \dots

Δ u_{14} = ∣ 0.40647 \dots - 0.77000 \dots ∣ = 0.36352 \dots

Δ u_{14} = 0.36352 \dots

Cannot find solution

The solution is

u \approx - 1.06603 \dots

Substitute back

u = cos (x)

cos (x) \approx - 1.06603 \dots

cos (x) \approx - 1.06603 \dots

cos (x) = - 1.06603 \dots, 0 \leq x < 2 π :

No Solution

cos (x) = - 1.06603 \dots, 0 \leq x < 2 π

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

No Solution for x \in R

Find the undefined points:

x = \frac{π}{2}, x = \frac{3 π}{2}, x = 0, x = π

Find the zeros of the denominator

cos (x) sin (x) = 0

Solving each part separately

cos (x) = 0 or sin (x) = 0

cos (x) = 0, 0 \leq x < 2 π : x = \frac{π}{2}, x = \frac{3 π}{2}

cos (x) = 0, 0 \leq x < 2 π

General solutions for

cos (x) = 0

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}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

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

Solutions for the range

0 \leq x < 2 π

x = \frac{π}{2}, x = \frac{3 π}{2}

sin (x) = 0, 0 \leq x < 2 π : x = 0, x = π

sin (x) = 0, 0 \leq x < 2 π

General solutions for

sin (x) = 0

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

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

Solutions for the range

0 \leq x < 2 π

x = 0, x = π

Combine all the solutions

x = \frac{π}{2}, x = \frac{3 π}{2}, x = 0, x = π

0, \frac{π}{2}, π, \frac{3 π}{2}

Identify the intervals

0 < x < \frac{π}{2}, \frac{π}{2} < x < π, π < x < \frac{3 π}{2}, \frac{3 π}{2} < x < 2 π

Summarize in a table:

3 sin^{2} (x) + cos^{2} (x) - 5 sin^{2} (x) cos (x) cos (x) sin (x) \frac{3 s i n ^{2} ( x ) + c o s ^{2} ( x ) - 5 s i n ^{2} ( x ) c o s ( x )}{c o s ( x ) s i n ( x )} x = 0 + + 0 Undefined 0 < x < \frac{π}{2} + + + + x = \frac{π}{2} + 0 + Undefined \frac{π}{2} < x < π + - + - x = π + - 0 Undefined π < x < \frac{3 π}{2} + - - + x = \frac{3 π}{2} + 0 - Undefined \frac{3 π}{2} < x < 2 π + + - - x = 2 π + + 0 Undefined

Identify the intervals that satisfy the required condition:

< 0

\frac{π}{2} < x < π or \frac{3 π}{2} < x < 2 π

Apply the periodicity of

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

\frac{π}{2} + 2 πn < x < π + 2 πn or \frac{3 π}{2} + 2 πn < x < 2 π + 2 πn

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