What is the general solution for csc^2(x)-3=6tan(x) ?

The general solution for csc^2(x)-3=6tan(x) is x=0.43008…+pin

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csc^2(x)-3=6tan(x)

Solution

csc^{2} (x) - 3 = 6 tan (x)

Solution

x = 0.43008 \dots + πn

Degrees

x = 24.64185 \dots^{\circ} + 18 0^{\circ} n

Solution steps

csc^{2} (x) - 3 = 6 tan (x)

Square both sides

(csc^{2} (x) - 3)^{2} = (6 tan (x))^{2}

Subtract

(6 tan (x))^{2}

from both sides

(csc^{2} (x) - 3)^{2} - 36 tan^{2} (x) = 0

Rewrite using trig identities

(- 3 + csc^{2} (x))^{2} - 36 tan^{2} (x)

Use the Pythagorean identity:

csc^{2} (x) = 1 + cot^{2} (x)

= (- 3 + 1 + cot^{2} (x))^{2} - 36 tan^{2} (x)

Simplify

= (cot^{2} (x) - 2)^{2} - 36 tan^{2} (x)

(- 2 + cot^{2} (x))^{2} - 36 tan^{2} (x) = 0

Factor

(- 2 + cot^{2} (x))^{2} - 36 tan^{2} (x) : (- 2 + cot^{2} (x) + 6 tan (x)) (- 2 + cot^{2} (x) - 6 tan (x))

(- 2 + cot^{2} (x))^{2} - 36 tan^{2} (x)

Rewrite

(- 2 + cot^{2} (x))^{2} - 36 tan^{2} (x)

(- 2 + cot^{2} (x))^{2} - (6 tan (x))^{2}

(- 2 + cot^{2} (x))^{2} - 36 tan^{2} (x)

Rewrite

36

6^{2}

= (- 2 + cot^{2} (x))^{2} - 6^{2} tan^{2} (x)

Apply exponent rule:

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

6^{2} tan^{2} (x) = (6 tan (x))^{2}

= (- 2 + cot^{2} (x))^{2} - (6 tan (x))^{2}

= (- 2 + cot^{2} (x))^{2} - (6 tan (x))^{2}

Apply Difference of Two Squares Formula:

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

(- 2 + cot^{2} (x))^{2} - (6 tan (x))^{2} = ((- 2 + cot^{2} (x)) + 6 tan (x)) ((- 2 + cot^{2} (x)) - 6 tan (x))

= ((- 2 + cot^{2} (x)) + 6 tan (x)) ((- 2 + cot^{2} (x)) - 6 tan (x))

Refine

= (cot^{2} (x) + 6 tan (x) - 2) (cot^{2} (x) - 6 tan (x) - 2)

(- 2 + cot^{2} (x) + 6 tan (x)) (- 2 + cot^{2} (x) - 6 tan (x)) = 0

Solving each part separately

- 2 + cot^{2} (x) + 6 tan (x) = 0 or - 2 + cot^{2} (x) - 6 tan (x) = 0

- 2 + cot^{2} (x) + 6 tan (x) = 0 : x = arccot (- 2.17998 \dots) + πn

- 2 + cot^{2} (x) + 6 tan (x) = 0

Rewrite using trig identities

- 2 + cot^{2} (x) + 6 tan (x)

Use the basic trigonometric identity:

tan (x) = \frac{1}{cot ( x )}

= - 2 + cot^{2} (x) + 6 \cdot \frac{1}{cot ( x )}

6 \cdot \frac{1}{cot ( x )} = \frac{6}{cot ( x )}

6 \cdot \frac{1}{cot ( x )}

Multiply fractions:

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

= \frac{1 \cdot 6}{cot ( x )}

Multiply the numbers:

1 \cdot 6 = 6

= \frac{6}{cot ( x )}

= - 2 + cot^{2} (x) + \frac{6}{cot ( x )}

- 2 + cot^{2} (x) + \frac{6}{cot ( x )} = 0

Solve by substitution

- 2 + cot^{2} (x) + \frac{6}{cot ( x )} = 0

Let:

cot (x) = u

- 2 + u^{2} + \frac{6}{u} = 0

- 2 + u^{2} + \frac{6}{u} = 0 : u \approx - 2.17998 \dots

- 2 + u^{2} + \frac{6}{u} = 0

Multiply both sides by

u

- 2 + u^{2} + \frac{6}{u} = 0

Multiply both sides by

u

- 2 u + u^{2} u + \frac{6}{u} u = 0 \cdot u

Simplify

- 2 u + u^{2} u + \frac{6}{u} u = 0 \cdot u

Simplify

u^{2} u : u^{3}

u^{2} u

Apply exponent rule:

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

u^{2} u = u^{2 + 1}

= u^{2 + 1}

Add the numbers:

2 + 1 = 3

= u^{3}

Simplify

\frac{6}{u} u : 6

\frac{6}{u} u

Multiply fractions:

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

= \frac{6 u}{u}

Cancel the common factor:

u

= 6

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

- 2 u + u^{3} + 6 = 0

- 2 u + u^{3} + 6 = 0

- 2 u + u^{3} + 6 = 0

Solve

- 2 u + u^{3} + 6 = 0 : u \approx - 2.17998 \dots

- 2 u + u^{3} + 6 = 0

Write in the standard form

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

u^{3} - 2 u + 6 = 0

Find one solution for

u^{3} - 2 u + 6 = 0

using Newton-Raphson:

u \approx - 2.17998 \dots

u^{3} - 2 u + 6 = 0

Newton-Raphson Approximation Definition

f (u) = u^{3} - 2 u + 6

Find

f^{^{'}} (u) : 3 u^{2} - 2

\frac{d}{d u} (u^{3} - 2 u + 6)

Apply the Sum/Difference Rule:

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

= \frac{d}{d u} (u^{3}) - \frac{d}{d u} (2 u) + \frac{d}{d u} (6)

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

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

Apply the Power Rule:

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

= 3 u^{3 - 1}

Simplify

= 3 u^{2}

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

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

Take the constant out:

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

= 2 \frac{d u}{d u}

Apply the common derivative:

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

= 2 \cdot 1

Simplify

= 2

\frac{d}{d u} (6) = 0

\frac{d}{d u} (6)

Derivative of a constant:

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

= 0

= 3 u^{2} - 2 + 0

Simplify

= 3 u^{2} - 2

Let

u_{0} = 3

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 1.92 : Δ u_{1} = 1.08

f (u_{0}) = 3^{3} - 2 \cdot 3 + 6 = 27

f^{^{'}} (u_{0}) = 3 \cdot 3^{2} - 2 = 25

u_{1} = 1.92

Δ u_{1} = ∣ 1.92 - 3 ∣ = 1.08

Δ u_{1} = 1.08

u_{2} = 0.90027 \dots : Δ u_{2} = 1.01972 \dots

f (u_{1}) = 1.9 2^{3} - 2 \cdot 1.92 + 6 = 9.237888

f^{^{'}} (u_{1}) = 3 \cdot 1.9 2^{2} - 2 = 9.0592

u_{2} = 0.90027 \dots

Δ u_{2} = ∣ 0.90027 \dots - 1.92 ∣ = 1.01972 \dots

Δ u_{2} = 1.01972 \dots

u_{3} = - 10.52325 \dots : Δ u_{3} = 11.42353 \dots

f (u_{2}) = 0.90027 \dots^{3} - 2 \cdot 0.90027 \dots + 6 = 4.92911 \dots

f^{^{'}} (u_{2}) = 3 \cdot 0.90027 \dots^{2} - 2 = 0.43148 \dots

u_{3} = - 10.52325 \dots

Δ u_{3} = ∣ - 10.52325 \dots - 0.90027 \dots ∣ = 11.42353 \dots

Δ u_{3} = 11.42353 \dots

u_{4} = - 7.07616 \dots : Δ u_{4} = 3.44709 \dots

f (u_{3}) = (- 10.52325 \dots)^{3} - 2 (- 10.52325 \dots) + 6 = - 1138.28856 \dots

f^{^{'}} (u_{3}) = 3 (- 10.52325 \dots)^{2} - 2 = 330.21696 \dots

u_{4} = - 7.07616 \dots

Δ u_{4} = ∣ - 7.07616 \dots - (- 10.52325 \dots) ∣ = 3.44709 \dots

Δ u_{4} = 3.44709 \dots

u_{5} = - 4.82158 \dots : Δ u_{5} = 2.25458 \dots

f (u_{4}) = (- 7.07616 \dots)^{3} - 2 (- 7.07616 \dots) + 6 = - 334.16639 \dots

f^{^{'}} (u_{4}) = 3 (- 7.07616 \dots)^{2} - 2 = 148.21639 \dots

u_{5} = - 4.82158 \dots

Δ u_{5} = ∣ - 4.82158 \dots - (- 7.07616 \dots) ∣ = 2.25458 \dots

Δ u_{5} = 2.25458 \dots

u_{6} = - 3.39785 \dots : Δ u_{6} = 1.42372 \dots

f (u_{5}) = (- 4.82158 \dots)^{3} - 2 (- 4.82158 \dots) + 6 = - 96.44728 \dots

f^{^{'}} (u_{5}) = 3 (- 4.82158 \dots)^{2} - 2 = 67.74295 \dots

u_{6} = - 3.39785 \dots

Δ u_{6} = ∣ - 3.39785 \dots - (- 4.82158 \dots) ∣ = 1.42372 \dots

Δ u_{6} = 1.42372 \dots

u_{7} = - 2.58789 \dots : Δ u_{7} = 0.80995 \dots

f (u_{6}) = (- 3.39785 \dots)^{3} - 2 (- 3.39785 \dots) + 6 = - 26.43402 \dots

f^{^{'}} (u_{6}) = 3 (- 3.39785 \dots)^{2} - 2 = 32.63630 \dots

u_{7} = - 2.58789 \dots

Δ u_{7} = ∣ - 2.58789 \dots - (- 3.39785 \dots) ∣ = 0.80995 \dots

Δ u_{7} = 0.80995 \dots

u_{8} = - 2.24763 \dots : Δ u_{8} = 0.34026 \dots

f (u_{7}) = (- 2.58789 \dots)^{3} - 2 (- 2.58789 \dots) + 6 = - 6.15594 \dots

f^{^{'}} (u_{7}) = 3 (- 2.58789 \dots)^{2} - 2 = 18.09167 \dots

u_{8} = - 2.24763 \dots

Δ u_{8} = ∣ - 2.24763 \dots - (- 2.58789 \dots) ∣ = 0.34026 \dots

Δ u_{8} = 0.34026 \dots

u_{9} = - 2.18230 \dots : Δ u_{9} = 0.06533 \dots

f (u_{8}) = (- 2.24763 \dots)^{3} - 2 (- 2.24763 \dots) + 6 = - 0.85948 \dots

f^{^{'}} (u_{8}) = 3 (- 2.24763 \dots)^{2} - 2 = 13.15559 \dots

u_{9} = - 2.18230 \dots

Δ u_{9} = ∣ - 2.18230 \dots - (- 2.24763 \dots) ∣ = 0.06533 \dots

Δ u_{9} = 0.06533 \dots

u_{10} = - 2.17998 \dots : Δ u_{10} = 0.00231 \dots

f (u_{9}) = (- 2.18230 \dots)^{3} - 2 (- 2.18230 \dots) + 6 = - 0.02850 \dots

f^{^{'}} (u_{9}) = 3 (- 2.18230 \dots)^{2} - 2 = 12.28734 \dots

u_{10} = - 2.17998 \dots

Δ u_{10} = ∣ - 2.17998 \dots - (- 2.18230 \dots) ∣ = 0.00231 \dots

Δ u_{10} = 0.00231 \dots

u_{11} = - 2.17998 \dots : Δ u_{11} = 2.87294 E - 6

f (u_{10}) = (- 2.17998 \dots)^{3} - 2 (- 2.17998 \dots) + 6 = - 0.00003 \dots

f^{^{'}} (u_{10}) = 3 (- 2.17998 \dots)^{2} - 2 = 12.25699 \dots

u_{11} = - 2.17998 \dots

Δ u_{11} = ∣ - 2.17998 \dots - (- 2.17998 \dots) ∣ = 2.87294 E - 6

Δ u_{11} = 2.87294 E - 6

u_{12} = - 2.17998 \dots : Δ u_{12} = 4.4041 E - 12

f (u_{11}) = (- 2.17998 \dots)^{3} - 2 (- 2.17998 \dots) + 6 = - 5.39808 E - 11

f^{^{'}} (u_{11}) = 3 (- 2.17998 \dots)^{2} - 2 = 12.25695 \dots

u_{12} = - 2.17998 \dots

Δ u_{12} = ∣ - 2.17998 \dots - (- 2.17998 \dots) ∣ = 4.4041 E - 12

Δ u_{12} = 4.4041 E - 12

u \approx - 2.17998 \dots

Apply long division:

\frac{u ^{3} - 2 u + 6}{u + 2.17998 \dots} = u^{2} - 2.17998 \dots u + 2.75231 \dots

u^{2} - 2.17998 \dots u + 2.75231 \dots \approx 0

Find one solution for

u^{2} - 2.17998 \dots u + 2.75231 \dots = 0

using Newton-Raphson:

No Solution for

u \in R

u^{2} - 2.17998 \dots u + 2.75231 \dots = 0

Newton-Raphson Approximation Definition

f (u) = u^{2} - 2.17998 \dots u + 2.75231 \dots

Find

f^{^{'}} (u) : 2 u - 2.17998 \dots

\frac{d}{d u} (u^{2} - 2.17998 \dots u + 2.75231 \dots)

Apply the Sum/Difference Rule:

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

= \frac{d}{d u} (u^{2}) - \frac{d}{d u} (2.17998 \dots u) + \frac{d}{d u} (2.75231 \dots)

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

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

Apply the Power Rule:

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

= 2 u^{2 - 1}

Simplify

= 2 u

\frac{d}{d u} (2.17998 \dots u) = 2.17998 \dots

\frac{d}{d u} (2.17998 \dots u)

Take the constant out:

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

= 2.17998 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 2.17998 \dots \cdot 1

Simplify

= 2.17998 \dots

\frac{d}{d u} (2.75231 \dots) = 0

\frac{d}{d u} (2.75231 \dots)

Derivative of a constant:

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

= 0

= 2 u - 2.17998 \dots + 0

Simplify

= 2 u - 2.17998 \dots

Let

u_{0} = 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 9.73612 \dots : Δ u_{1} = 8.73612 \dots

f (u_{0}) = 1^{2} - 2.17998 \dots \cdot 1 + 2.75231 \dots = 1.57233 \dots

f^{^{'}} (u_{0}) = 2 \cdot 1 - 2.17998 \dots = - 0.17998 \dots

u_{1} = 9.73612 \dots

Δ u_{1} = ∣ 9.73612 \dots - 1 ∣ = 8.73612 \dots

Δ u_{1} = 8.73612 \dots

u_{2} = 5.32259 \dots : Δ u_{2} = 4.41352 \dots

f (u_{1}) = 9.73612 \dots^{2} - 2.17998 \dots \cdot 9.73612 \dots + 2.75231 \dots = 76.31980 \dots

f^{^{'}} (u_{1}) = 2 \cdot 9.73612 \dots - 2.17998 \dots = 17.29226 \dots

u_{2} = 5.32259 \dots

Δ u_{2} = ∣ 5.32259 \dots - 9.73612 \dots ∣ = 4.41352 \dots

Δ u_{2} = 4.41352 \dots

u_{3} = 3.02150 \dots : Δ u_{3} = 2.30108 \dots

f (u_{2}) = 5.32259 \dots^{2} - 2.17998 \dots \cdot 5.32259 \dots + 2.75231 \dots = 19.47919 \dots

f^{^{'}} (u_{2}) = 2 \cdot 5.32259 \dots - 2.17998 \dots = 8.46521 \dots

u_{3} = 3.02150 \dots

Δ u_{3} = ∣ 3.02150 \dots - 5.32259 \dots ∣ = 2.30108 \dots

Δ u_{3} = 2.30108 \dots

u_{4} = 1.65082 \dots : Δ u_{4} = 1.37068 \dots

f (u_{3}) = 3.02150 \dots^{2} - 2.17998 \dots \cdot 3.02150 \dots + 2.75231 \dots = 5.29500 \dots

f^{^{'}} (u_{3}) = 2 \cdot 3.02150 \dots - 2.17998 \dots = 3.86303 \dots

u_{4} = 1.65082 \dots

Δ u_{4} = ∣ 1.65082 \dots - 3.02150 \dots ∣ = 1.37068 \dots

Δ u_{4} = 1.37068 \dots

u_{5} = - 0.02415 \dots : Δ u_{5} = 1.67497 \dots

f (u_{4}) = 1.65082 \dots^{2} - 2.17998 \dots \cdot 1.65082 \dots + 2.75231 \dots = 1.87877 \dots

f^{^{'}} (u_{4}) = 2 \cdot 1.65082 \dots - 2.17998 \dots = 1.12167 \dots

u_{5} = - 0.02415 \dots

Δ u_{5} = ∣ - 0.02415 \dots - 1.65082 \dots ∣ = 1.67497 \dots

Δ u_{5} = 1.67497 \dots

u_{6} = 1.23490 \dots : Δ u_{6} = 1.25906 \dots

f (u_{5}) = (- 0.02415 \dots)^{2} - 2.17998 \dots (- 0.02415 \dots) + 2.75231 \dots = 2.80555 \dots

f^{^{'}} (u_{5}) = 2 (- 0.02415 \dots) - 2.17998 \dots = - 2.22828 \dots

u_{6} = 1.23490 \dots

Δ u_{6} = ∣ 1.23490 \dots - (- 0.02415 \dots) ∣ = 1.25906 \dots

Δ u_{6} = 1.25906 \dots

u_{7} = - 4.23449 \dots : Δ u_{7} = 5.46939 \dots

f (u_{6}) = 1.23490 \dots^{2} - 2.17998 \dots \cdot 1.23490 \dots + 2.75231 \dots = 1.58523 \dots

f^{^{'}} (u_{6}) = 2 \cdot 1.23490 \dots - 2.17998 \dots = 0.28983 \dots

u_{7} = - 4.23449 \dots

Δ u_{7} = ∣ - 4.23449 \dots - 1.23490 \dots ∣ = 5.46939 \dots

Δ u_{7} = 5.46939 \dots

u_{8} = - 1.42535 \dots : Δ u_{8} = 2.80913 \dots

f (u_{7}) = (- 4.23449 \dots)^{2} - 2.17998 \dots (- 4.23449 \dots) + 2.75231 \dots = 29.91433 \dots

f^{^{'}} (u_{7}) = 2 (- 4.23449 \dots) - 2.17998 \dots = - 10.64896 \dots

u_{8} = - 1.42535 \dots

Δ u_{8} = ∣ - 1.42535 \dots - (- 4.23449 \dots) ∣ = 2.80913 \dots

Δ u_{8} = 2.80913 \dots

u_{9} = 0.14325 \dots : Δ u_{9} = 1.56861 \dots

f (u_{8}) = (- 1.42535 \dots)^{2} - 2.17998 \dots (- 1.42535 \dots) + 2.75231 \dots = 7.89122 \dots

f^{^{'}} (u_{8}) = 2 (- 1.42535 \dots) - 2.17998 \dots = - 5.03069 \dots

u_{9} = 0.14325 \dots

Δ u_{9} = ∣ 0.14325 \dots - (- 1.42535 \dots) ∣ = 1.56861 \dots

Δ u_{9} = 1.56861 \dots

u_{10} = 1.44274 \dots : Δ u_{10} = 1.29948 \dots

f (u_{9}) = 0.14325 \dots^{2} - 2.17998 \dots \cdot 0.14325 \dots + 2.75231 \dots = 2.46054 \dots

f^{^{'}} (u_{9}) = 2 \cdot 0.14325 \dots - 2.17998 \dots = - 1.89347 \dots

u_{10} = 1.44274 \dots

Δ u_{10} = ∣ 1.44274 \dots - 0.14325 \dots ∣ = 1.29948 \dots

Δ u_{10} = 1.29948 \dots

u_{11} = - 0.95081 \dots : Δ u_{11} = 2.39356 \dots

f (u_{10}) = 1.44274 \dots^{2} - 2.17998 \dots \cdot 1.44274 \dots + 2.75231 \dots = 1.68867 \dots

f^{^{'}} (u_{10}) = 2 \cdot 1.44274 \dots - 2.17998 \dots = 0.70550 \dots

u_{11} = - 0.95081 \dots

Δ u_{11} = ∣ - 0.95081 \dots - 1.44274 \dots ∣ = 2.39356 \dots

Δ u_{11} = 2.39356 \dots

u_{12} = 0.45282 \dots : Δ u_{12} = 1.40364 \dots

f (u_{11}) = (- 0.95081 \dots)^{2} - 2.17998 \dots (- 0.95081 \dots) + 2.75231 \dots = 5.72913 \dots

f^{^{'}} (u_{11}) = 2 (- 0.95081 \dots) - 2.17998 \dots = - 4.08161 \dots

u_{12} = 0.45282 \dots

Δ u_{12} = ∣ 0.45282 \dots - (- 0.95081 \dots) ∣ = 1.40364 \dots

Δ u_{12} = 1.40364 \dots

u_{13} = 1.99891 \dots : Δ u_{13} = 1.54608 \dots

f (u_{12}) = 0.45282 \dots^{2} - 2.17998 \dots \cdot 0.45282 \dots + 2.75231 \dots = 1.97021 \dots

f^{^{'}} (u_{12}) = 2 \cdot 0.45282 \dots - 2.17998 \dots = - 1.27432 \dots

u_{13} = 1.99891 \dots

Δ u_{13} = ∣ 1.99891 \dots - 0.45282 \dots ∣ = 1.54608 \dots

Δ u_{13} = 1.54608 \dots

Cannot find solution

The solution is

u \approx - 2.17998 \dots

u \approx - 2.17998 \dots

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 2 + u^{2} + \frac{6}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u \approx - 2.17998 \dots

Substitute back

u = cot (x)

cot (x) \approx - 2.17998 \dots

cot (x) \approx - 2.17998 \dots

cot (x) = - 2.17998 \dots : x = arccot (- 2.17998 \dots) + πn

cot (x) = - 2.17998 \dots

Apply trig inverse properties

cot (x) = - 2.17998 \dots

General solutions for

cot (x) = - 2.17998 \dots

cot (x) = - a \Rightarrow x = arccot (- a) + πn

x = arccot (- 2.17998 \dots) + πn

x = arccot (- 2.17998 \dots) + πn

Combine all the solutions

x = arccot (- 2.17998 \dots) + πn

- 2 + cot^{2} (x) - 6 tan (x) = 0 : x = arccot (2.17998 \dots) + πn

- 2 + cot^{2} (x) - 6 tan (x) = 0

Rewrite using trig identities

- 2 + cot^{2} (x) - 6 tan (x)

Use the basic trigonometric identity:

tan (x) = \frac{1}{cot ( x )}

= - 2 + cot^{2} (x) - 6 \cdot \frac{1}{cot ( x )}

6 \cdot \frac{1}{cot ( x )} = \frac{6}{cot ( x )}

6 \cdot \frac{1}{cot ( x )}

Multiply fractions:

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

= \frac{1 \cdot 6}{cot ( x )}

Multiply the numbers:

1 \cdot 6 = 6

= \frac{6}{cot ( x )}

= - 2 + cot^{2} (x) - \frac{6}{cot ( x )}

- 2 + cot^{2} (x) - \frac{6}{cot ( x )} = 0

Solve by substitution

- 2 + cot^{2} (x) - \frac{6}{cot ( x )} = 0

Let:

cot (x) = u

- 2 + u^{2} - \frac{6}{u} = 0

- 2 + u^{2} - \frac{6}{u} = 0 : u \approx 2.17998 \dots

- 2 + u^{2} - \frac{6}{u} = 0

Multiply both sides by

u

- 2 + u^{2} - \frac{6}{u} = 0

Multiply both sides by

u

- 2 u + u^{2} u - \frac{6}{u} u = 0 \cdot u

Simplify

- 2 u + u^{2} u - \frac{6}{u} u = 0 \cdot u

Simplify

u^{2} u : u^{3}

u^{2} u

Apply exponent rule:

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

u^{2} u = u^{2 + 1}

= u^{2 + 1}

Add the numbers:

2 + 1 = 3

= u^{3}

Simplify

- \frac{6}{u} u : - 6

- \frac{6}{u} u

Multiply fractions:

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

= - \frac{6 u}{u}

Cancel the common factor:

u

= - 6

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

- 2 u + u^{3} - 6 = 0

- 2 u + u^{3} - 6 = 0

- 2 u + u^{3} - 6 = 0

Solve

- 2 u + u^{3} - 6 = 0 : u \approx 2.17998 \dots

- 2 u + u^{3} - 6 = 0

Write in the standard form

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

u^{3} - 2 u - 6 = 0

Find one solution for

u^{3} - 2 u - 6 = 0

using Newton-Raphson:

u \approx 2.17998 \dots

u^{3} - 2 u - 6 = 0

Newton-Raphson Approximation Definition

f (u) = u^{3} - 2 u - 6

Find

f^{^{'}} (u) : 3 u^{2} - 2

\frac{d}{d u} (u^{3} - 2 u - 6)

Apply the Sum/Difference Rule:

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

= \frac{d}{d u} (u^{3}) - \frac{d}{d u} (2 u) - \frac{d}{d u} (6)

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

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

Apply the Power Rule:

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

= 3 u^{3 - 1}

Simplify

= 3 u^{2}

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

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

Take the constant out:

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

= 2 \frac{d u}{d u}

Apply the common derivative:

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

= 2 \cdot 1

Simplify

= 2

\frac{d}{d u} (6) = 0

\frac{d}{d u} (6)

Derivative of a constant:

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

= 0

= 3 u^{2} - 2 - 0

Simplify

= 3 u^{2} - 2

Let

u_{0} = - 3

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 1.92 : Δ u_{1} = 1.08

f (u_{0}) = (- 3)^{3} - 2 (- 3) - 6 = - 27

f^{^{'}} (u_{0}) = 3 (- 3)^{2} - 2 = 25

u_{1} = - 1.92

Δ u_{1} = ∣ - 1.92 - (- 3) ∣ = 1.08

Δ u_{1} = 1.08

u_{2} = - 0.90027 \dots : Δ u_{2} = 1.01972 \dots

f (u_{1}) = (- 1.92)^{3} - 2 (- 1.92) - 6 = - 9.237888

f^{^{'}} (u_{1}) = 3 (- 1.92)^{2} - 2 = 9.0592

u_{2} = - 0.90027 \dots

Δ u_{2} = ∣ - 0.90027 \dots - (- 1.92) ∣ = 1.01972 \dots

Δ u_{2} = 1.01972 \dots

u_{3} = 10.52325 \dots : Δ u_{3} = 11.42353 \dots

f (u_{2}) = (- 0.90027 \dots)^{3} - 2 (- 0.90027 \dots) - 6 = - 4.92911 \dots

f^{^{'}} (u_{2}) = 3 (- 0.90027 \dots)^{2} - 2 = 0.43148 \dots

u_{3} = 10.52325 \dots

Δ u_{3} = ∣ 10.52325 \dots - (- 0.90027 \dots) ∣ = 11.42353 \dots

Δ u_{3} = 11.42353 \dots

u_{4} = 7.07616 \dots : Δ u_{4} = 3.44709 \dots

f (u_{3}) = 10.52325 \dots^{3} - 2 \cdot 10.52325 \dots - 6 = 1138.28856 \dots

f^{^{'}} (u_{3}) = 3 \cdot 10.52325 \dots^{2} - 2 = 330.21696 \dots

u_{4} = 7.07616 \dots

Δ u_{4} = ∣ 7.07616 \dots - 10.52325 \dots ∣ = 3.44709 \dots

Δ u_{4} = 3.44709 \dots

u_{5} = 4.82158 \dots : Δ u_{5} = 2.25458 \dots

f (u_{4}) = 7.07616 \dots^{3} - 2 \cdot 7.07616 \dots - 6 = 334.16639 \dots

f^{^{'}} (u_{4}) = 3 \cdot 7.07616 \dots^{2} - 2 = 148.21639 \dots

u_{5} = 4.82158 \dots

Δ u_{5} = ∣ 4.82158 \dots - 7.07616 \dots ∣ = 2.25458 \dots

Δ u_{5} = 2.25458 \dots

u_{6} = 3.39785 \dots : Δ u_{6} = 1.42372 \dots

f (u_{5}) = 4.82158 \dots^{3} - 2 \cdot 4.82158 \dots - 6 = 96.44728 \dots

f^{^{'}} (u_{5}) = 3 \cdot 4.82158 \dots^{2} - 2 = 67.74295 \dots

u_{6} = 3.39785 \dots

Δ u_{6} = ∣ 3.39785 \dots - 4.82158 \dots ∣ = 1.42372 \dots

Δ u_{6} = 1.42372 \dots

u_{7} = 2.58789 \dots : Δ u_{7} = 0.80995 \dots

f (u_{6}) = 3.39785 \dots^{3} - 2 \cdot 3.39785 \dots - 6 = 26.43402 \dots

f^{^{'}} (u_{6}) = 3 \cdot 3.39785 \dots^{2} - 2 = 32.63630 \dots

u_{7} = 2.58789 \dots

Δ u_{7} = ∣ 2.58789 \dots - 3.39785 \dots ∣ = 0.80995 \dots

Δ u_{7} = 0.80995 \dots

u_{8} = 2.24763 \dots : Δ u_{8} = 0.34026 \dots

f (u_{7}) = 2.58789 \dots^{3} - 2 \cdot 2.58789 \dots - 6 = 6.15594 \dots

f^{^{'}} (u_{7}) = 3 \cdot 2.58789 \dots^{2} - 2 = 18.09167 \dots

u_{8} = 2.24763 \dots

Δ u_{8} = ∣ 2.24763 \dots - 2.58789 \dots ∣ = 0.34026 \dots

Δ u_{8} = 0.34026 \dots

u_{9} = 2.18230 \dots : Δ u_{9} = 0.06533 \dots

f (u_{8}) = 2.24763 \dots^{3} - 2 \cdot 2.24763 \dots - 6 = 0.85948 \dots

f^{^{'}} (u_{8}) = 3 \cdot 2.24763 \dots^{2} - 2 = 13.15559 \dots

u_{9} = 2.18230 \dots

Δ u_{9} = ∣ 2.18230 \dots - 2.24763 \dots ∣ = 0.06533 \dots

Δ u_{9} = 0.06533 \dots

u_{10} = 2.17998 \dots : Δ u_{10} = 0.00231 \dots

f (u_{9}) = 2.18230 \dots^{3} - 2 \cdot 2.18230 \dots - 6 = 0.02850 \dots

f^{^{'}} (u_{9}) = 3 \cdot 2.18230 \dots^{2} - 2 = 12.28734 \dots

u_{10} = 2.17998 \dots

Δ u_{10} = ∣ 2.17998 \dots - 2.18230 \dots ∣ = 0.00231 \dots

Δ u_{10} = 0.00231 \dots

u_{11} = 2.17998 \dots : Δ u_{11} = 2.87294 E - 6

f (u_{10}) = 2.17998 \dots^{3} - 2 \cdot 2.17998 \dots - 6 = 0.00003 \dots

f^{^{'}} (u_{10}) = 3 \cdot 2.17998 \dots^{2} - 2 = 12.25699 \dots

u_{11} = 2.17998 \dots

Δ u_{11} = ∣ 2.17998 \dots - 2.17998 \dots ∣ = 2.87294 E - 6

Δ u_{11} = 2.87294 E - 6

u_{12} = 2.17998 \dots : Δ u_{12} = 4.4041 E - 12

f (u_{11}) = 2.17998 \dots^{3} - 2 \cdot 2.17998 \dots - 6 = 5.39808 E - 11

f^{^{'}} (u_{11}) = 3 \cdot 2.17998 \dots^{2} - 2 = 12.25695 \dots

u_{12} = 2.17998 \dots

Δ u_{12} = ∣ 2.17998 \dots - 2.17998 \dots ∣ = 4.4041 E - 12

Δ u_{12} = 4.4041 E - 12

u \approx 2.17998 \dots

Apply long division:

\frac{u ^{3} - 2 u - 6}{u - 2.17998 \dots} = u^{2} + 2.17998 \dots u + 2.75231 \dots

u^{2} + 2.17998 \dots u + 2.75231 \dots \approx 0

Find one solution for

u^{2} + 2.17998 \dots u + 2.75231 \dots = 0

using Newton-Raphson:

No Solution for

u \in R

u^{2} + 2.17998 \dots u + 2.75231 \dots = 0

Newton-Raphson Approximation Definition

f (u) = u^{2} + 2.17998 \dots u + 2.75231 \dots

Find

f^{^{'}} (u) : 2 u + 2.17998 \dots

\frac{d}{d u} (u^{2} + 2.17998 \dots u + 2.75231 \dots)

Apply the Sum/Difference Rule:

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

= \frac{d}{d u} (u^{2}) + \frac{d}{d u} (2.17998 \dots u) + \frac{d}{d u} (2.75231 \dots)

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

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

Apply the Power Rule:

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

= 2 u^{2 - 1}

Simplify

= 2 u

\frac{d}{d u} (2.17998 \dots u) = 2.17998 \dots

\frac{d}{d u} (2.17998 \dots u)

Take the constant out:

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

= 2.17998 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 2.17998 \dots \cdot 1

Simplify

= 2.17998 \dots

\frac{d}{d u} (2.75231 \dots) = 0

\frac{d}{d u} (2.75231 \dots)

Derivative of a constant:

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

= 0

= 2 u + 2.17998 \dots + 0

Simplify

= 2 u + 2.17998 \dots

Let

u_{0} = - 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 9.73612 \dots : Δ u_{1} = 8.73612 \dots

f (u_{0}) = (- 1)^{2} + 2.17998 \dots (- 1) + 2.75231 \dots = 1.57233 \dots

f^{^{'}} (u_{0}) = 2 (- 1) + 2.17998 \dots = 0.17998 \dots

u_{1} = - 9.73612 \dots

Δ u_{1} = ∣ - 9.73612 \dots - (- 1) ∣ = 8.73612 \dots

Δ u_{1} = 8.73612 \dots

u_{2} = - 5.32259 \dots : Δ u_{2} = 4.41352 \dots

f (u_{1}) = (- 9.73612 \dots)^{2} + 2.17998 \dots (- 9.73612 \dots) + 2.75231 \dots = 76.31980 \dots

f^{^{'}} (u_{1}) = 2 (- 9.73612 \dots) + 2.17998 \dots = - 17.29226 \dots

u_{2} = - 5.32259 \dots

Δ u_{2} = ∣ - 5.32259 \dots - (- 9.73612 \dots) ∣ = 4.41352 \dots

Δ u_{2} = 4.41352 \dots

u_{3} = - 3.02150 \dots : Δ u_{3} = 2.30108 \dots

f (u_{2}) = (- 5.32259 \dots)^{2} + 2.17998 \dots (- 5.32259 \dots) + 2.75231 \dots = 19.47919 \dots

f^{^{'}} (u_{2}) = 2 (- 5.32259 \dots) + 2.17998 \dots = - 8.46521 \dots

u_{3} = - 3.02150 \dots

Δ u_{3} = ∣ - 3.02150 \dots - (- 5.32259 \dots) ∣ = 2.30108 \dots

Δ u_{3} = 2.30108 \dots

u_{4} = - 1.65082 \dots : Δ u_{4} = 1.37068 \dots

f (u_{3}) = (- 3.02150 \dots)^{2} + 2.17998 \dots (- 3.02150 \dots) + 2.75231 \dots = 5.29500 \dots

f^{^{'}} (u_{3}) = 2 (- 3.02150 \dots) + 2.17998 \dots = - 3.86303 \dots

u_{4} = - 1.65082 \dots

Δ u_{4} = ∣ - 1.65082 \dots - (- 3.02150 \dots) ∣ = 1.37068 \dots

Δ u_{4} = 1.37068 \dots

u_{5} = 0.02415 \dots : Δ u_{5} = 1.67497 \dots

f (u_{4}) = (- 1.65082 \dots)^{2} + 2.17998 \dots (- 1.65082 \dots) + 2.75231 \dots = 1.87877 \dots

f^{^{'}} (u_{4}) = 2 (- 1.65082 \dots) + 2.17998 \dots = - 1.12167 \dots

u_{5} = 0.02415 \dots

Δ u_{5} = ∣ 0.02415 \dots - (- 1.65082 \dots) ∣ = 1.67497 \dots

Δ u_{5} = 1.67497 \dots

u_{6} = - 1.23490 \dots : Δ u_{6} = 1.25906 \dots

f (u_{5}) = 0.02415 \dots^{2} + 2.17998 \dots \cdot 0.02415 \dots + 2.75231 \dots = 2.80555 \dots

f^{^{'}} (u_{5}) = 2 \cdot 0.02415 \dots + 2.17998 \dots = 2.22828 \dots

u_{6} = - 1.23490 \dots

Δ u_{6} = ∣ - 1.23490 \dots - 0.02415 \dots ∣ = 1.25906 \dots

Δ u_{6} = 1.25906 \dots

u_{7} = 4.23449 \dots : Δ u_{7} = 5.46939 \dots

f (u_{6}) = (- 1.23490 \dots)^{2} + 2.17998 \dots (- 1.23490 \dots) + 2.75231 \dots = 1.58523 \dots

f^{^{'}} (u_{6}) = 2 (- 1.23490 \dots) + 2.17998 \dots = - 0.28983 \dots

u_{7} = 4.23449 \dots

Δ u_{7} = ∣ 4.23449 \dots - (- 1.23490 \dots) ∣ = 5.46939 \dots

Δ u_{7} = 5.46939 \dots

u_{8} = 1.42535 \dots : Δ u_{8} = 2.80913 \dots

f (u_{7}) = 4.23449 \dots^{2} + 2.17998 \dots \cdot 4.23449 \dots + 2.75231 \dots = 29.91433 \dots

f^{^{'}} (u_{7}) = 2 \cdot 4.23449 \dots + 2.17998 \dots = 10.64896 \dots

u_{8} = 1.42535 \dots

Δ u_{8} = ∣ 1.42535 \dots - 4.23449 \dots ∣ = 2.80913 \dots

Δ u_{8} = 2.80913 \dots

u_{9} = - 0.14325 \dots : Δ u_{9} = 1.56861 \dots

f (u_{8}) = 1.42535 \dots^{2} + 2.17998 \dots \cdot 1.42535 \dots + 2.75231 \dots = 7.89122 \dots

f^{^{'}} (u_{8}) = 2 \cdot 1.42535 \dots + 2.17998 \dots = 5.03069 \dots

u_{9} = - 0.14325 \dots

Δ u_{9} = ∣ - 0.14325 \dots - 1.42535 \dots ∣ = 1.56861 \dots

Δ u_{9} = 1.56861 \dots

u_{10} = - 1.44274 \dots : Δ u_{10} = 1.29948 \dots

f (u_{9}) = (- 0.14325 \dots)^{2} + 2.17998 \dots (- 0.14325 \dots) + 2.75231 \dots = 2.46054 \dots

f^{^{'}} (u_{9}) = 2 (- 0.14325 \dots) + 2.17998 \dots = 1.89347 \dots

u_{10} = - 1.44274 \dots

Δ u_{10} = ∣ - 1.44274 \dots - (- 0.14325 \dots) ∣ = 1.29948 \dots

Δ u_{10} = 1.29948 \dots

u_{11} = 0.95081 \dots : Δ u_{11} = 2.39356 \dots

f (u_{10}) = (- 1.44274 \dots)^{2} + 2.17998 \dots (- 1.44274 \dots) + 2.75231 \dots = 1.68867 \dots

f^{^{'}} (u_{10}) = 2 (- 1.44274 \dots) + 2.17998 \dots = - 0.70550 \dots

u_{11} = 0.95081 \dots

Δ u_{11} = ∣ 0.95081 \dots - (- 1.44274 \dots) ∣ = 2.39356 \dots

Δ u_{11} = 2.39356 \dots

u_{12} = - 0.45282 \dots : Δ u_{12} = 1.40364 \dots

f (u_{11}) = 0.95081 \dots^{2} + 2.17998 \dots \cdot 0.95081 \dots + 2.75231 \dots = 5.72913 \dots

f^{^{'}} (u_{11}) = 2 \cdot 0.95081 \dots + 2.17998 \dots = 4.08161 \dots

u_{12} = - 0.45282 \dots

Δ u_{12} = ∣ - 0.45282 \dots - 0.95081 \dots ∣ = 1.40364 \dots

Δ u_{12} = 1.40364 \dots

u_{13} = - 1.99891 \dots : Δ u_{13} = 1.54608 \dots

f (u_{12}) = (- 0.45282 \dots)^{2} + 2.17998 \dots (- 0.45282 \dots) + 2.75231 \dots = 1.97021 \dots

f^{^{'}} (u_{12}) = 2 (- 0.45282 \dots) + 2.17998 \dots = 1.27432 \dots

u_{13} = - 1.99891 \dots

Δ u_{13} = ∣ - 1.99891 \dots - (- 0.45282 \dots) ∣ = 1.54608 \dots

Δ u_{13} = 1.54608 \dots

Cannot find solution

The solution is

u \approx 2.17998 \dots

u \approx 2.17998 \dots

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 2 + u^{2} - \frac{6}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u \approx 2.17998 \dots

Substitute back

u = cot (x)

cot (x) \approx 2.17998 \dots

cot (x) \approx 2.17998 \dots

cot (x) = 2.17998 \dots : x = arccot (2.17998 \dots) + πn

cot (x) = 2.17998 \dots

Apply trig inverse properties

cot (x) = 2.17998 \dots

General solutions for

cot (x) = 2.17998 \dots

cot (x) = a \Rightarrow x = arccot (a) + πn

x = arccot (2.17998 \dots) + πn

x = arccot (2.17998 \dots) + πn

Combine all the solutions

x = arccot (2.17998 \dots) + πn

Combine all the solutions

x = arccot (- 2.17998 \dots) + πn, x = arccot (2.17998 \dots) + πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

csc^{2} (x) - 3 = 6 tan (x)

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

Check the solution

arccot (- 2.17998 \dots) + πn :

False

arccot (- 2.17998 \dots) + πn

Plug in

n = 1

arccot (- 2.17998 \dots) + π 1

For

csc^{2} (x) - 3 = 6 tan (x)

plug in

x = arccot (- 2.17998 \dots) + π 1

csc^{2} (arccot (- 2.17998 \dots) + π 1) - 3 = 6 tan (arccot (- 2.17998 \dots) + π 1)

Refine

2.75231 \dots = - 2.75231 \dots

\Rightarrow False

Check the solution

arccot (2.17998 \dots) + πn :

True

arccot (2.17998 \dots) + πn

Plug in

n = 1

arccot (2.17998 \dots) + π 1

For

csc^{2} (x) - 3 = 6 tan (x)

plug in

x = arccot (2.17998 \dots) + π 1

csc^{2} (arccot (2.17998 \dots) + π 1) - 3 = 6 tan (arccot (2.17998 \dots) + π 1)

Refine

2.75231 \dots = 2.75231 \dots

\Rightarrow True

x = arccot (2.17998 \dots) + πn

Show solutions in decimal form

x = 0.43008 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for csc^2(x)-3=6tan(x) ?
The general solution for csc^2(x)-3=6tan(x) is x=0.43008…+pin