What is the general solution for sin(x)+2=tan(x) ?

The general solution for sin(x)+2=tan(x) is x=pi+0.88628…+2pin,x=1.24365…+2pin

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sin(x)+2=tan(x)

Solution

sin (x) + 2 = tan (x)

Solution

x = π + 0.88628 \dots + 2 πn, x = 1.24365 \dots + 2 πn

Degrees

x = 230.78052 \dots^{\circ} + 36 0^{\circ} n, x = 71.25627 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (x) + 2 = tan (x)

Square both sides

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

Subtract

tan^{2} (x)

from both sides

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

Rewrite using trig identities

(2 + sin (x))^{2} - tan^{2} (x)

Use the basic trigonometric identity:

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

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

Apply exponent rule:

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

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

Use the Pythagorean identity:

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

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

= (2 + sin (x))^{2} - \frac{sin ^{2} ( x )}{1 - sin ^{2} ( x )}

Simplify

(2 + sin (x))^{2} - \frac{sin ^{2} ( x )}{1 - sin ^{2} ( x )} : \frac{- 4 sin ^{2} ( x ) + 4 sin ( x ) - 4 sin ^{3} ( x ) - sin ^{4} ( x ) + 4}{1 - sin ^{2} ( x )}

(2 + sin (x))^{2} - \frac{sin ^{2} ( x )}{1 - sin ^{2} ( x )}

Convert element to fraction:

(sin (x) + 2)^{2} = \frac{( 2 + sin ( x ) ) ^{2} ( 1 - sin ^{2} ( x ) )}{1 - sin ^{2} ( x )}

= - \frac{sin ^{2} ( x )}{1 - sin ^{2} ( x )} + \frac{( 2 + sin ( x ) ) ^{2} ( 1 - sin ^{2} ( x ) )}{1 - sin ^{2} ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{- sin ^{2} ( x ) + ( 2 + sin ( x ) ) ^{2} ( 1 - sin ^{2} ( x ) )}{1 - sin ^{2} ( x )}

Expand

- sin^{2} (x) + (2 + sin (x))^{2} (1 - sin^{2} (x)) : - 4 sin^{2} (x) + 4 sin (x) - 4 sin^{3} (x) - sin^{4} (x) + 4

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

(2 + sin (x))^{2} : 4 + 4 sin (x) + sin^{2} (x)

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = 2, b = sin (x)

= 2^{2} + 2 \cdot 2 sin (x) + sin^{2} (x)

Simplify

2^{2} + 2 \cdot 2 sin (x) + sin^{2} (x) : 4 + 4 sin (x) + sin^{2} (x)

2^{2} + 2 \cdot 2 sin (x) + sin^{2} (x)

2^{2} = 4

= 4 + 2 \cdot 2 sin (x) + sin^{2} (x)

Multiply the numbers:

2 \cdot 2 = 4

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

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

= - sin^{2} (x) + (4 + 4 sin (x) + sin^{2} (x)) (1 - sin^{2} (x))

Expand

(4 + 4 sin (x) + sin^{2} (x)) (1 - sin^{2} (x)) : - 3 sin^{2} (x) + 4 sin (x) - 4 sin^{3} (x) - sin^{4} (x) + 4

(4 + 4 sin (x) + sin^{2} (x)) (1 - sin^{2} (x))

Distribute parentheses

= 4 \cdot 1 + 4 (- sin^{2} (x)) + 4 sin (x) \cdot 1 + 4 sin (x) (- sin^{2} (x)) + sin^{2} (x) \cdot 1 + sin^{2} (x) (- sin^{2} (x))

Apply minus-plus rules

+ (- a) = - a

= 4 \cdot 1 - 4 sin^{2} (x) + 4 \cdot 1 \cdot sin (x) - 4 sin^{2} (x) sin (x) + 1 \cdot sin^{2} (x) - sin^{2} (x) sin^{2} (x)

Simplify

4 \cdot 1 - 4 sin^{2} (x) + 4 \cdot 1 \cdot sin (x) - 4 sin^{2} (x) sin (x) + 1 \cdot sin^{2} (x) - sin^{2} (x) sin^{2} (x) : - 3 sin^{2} (x) + 4 sin (x) - 4 sin^{3} (x) - sin^{4} (x) + 4

4 \cdot 1 - 4 sin^{2} (x) + 4 \cdot 1 \cdot sin (x) - 4 sin^{2} (x) sin (x) + 1 \cdot sin^{2} (x) - sin^{2} (x) sin^{2} (x)

Group like terms

= - 4 sin^{2} (x) + 4 \cdot 1 \cdot sin (x) - 4 sin^{2} (x) sin (x) + 1 \cdot sin^{2} (x) - sin^{2} (x) sin^{2} (x) + 4 \cdot 1

Add similar elements:

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

= - 3 sin^{2} (x) + 4 \cdot 1 \cdot sin (x) - 4 sin^{2} (x) sin (x) - sin^{2} (x) sin^{2} (x) + 4 \cdot 1

4 \cdot 1 \cdot sin (x) = 4 sin (x)

4 \cdot 1 \cdot sin (x)

Multiply the numbers:

4 \cdot 1 = 4

= 4 sin (x)

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

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

Apply exponent rule:

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

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

= 4 sin^{2 + 1} (x)

Add the numbers:

2 + 1 = 3

= 4 sin^{3} (x)

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

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

Apply exponent rule:

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

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

= sin^{2 + 2} (x)

Add the numbers:

2 + 2 = 4

= sin^{4} (x)

4 \cdot 1 = 4

4 \cdot 1

Multiply the numbers:

4 \cdot 1 = 4

= 4

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

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

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

Add similar elements:

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

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

= \frac{- 4 sin ^{2} ( x ) + 4 sin ( x ) - 4 sin ^{3} ( x ) - sin ^{4} ( x ) + 4}{1 - sin ^{2} ( x )}

= \frac{- 4 sin ^{2} ( x ) + 4 sin ( x ) - 4 sin ^{3} ( x ) - sin ^{4} ( x ) + 4}{1 - sin ^{2} ( x )}

\frac{4 - sin ^{4} ( x ) + 4 sin ( x ) - 4 sin ^{2} ( x ) - 4 sin ^{3} ( x )}{1 - sin ^{2} ( x )} = 0

Solve by substitution

\frac{4 - sin ^{4} ( x ) + 4 sin ( x ) - 4 sin ^{2} ( x ) - 4 sin ^{3} ( x )}{1 - sin ^{2} ( x )} = 0

Let:

sin (x) = u

\frac{4 - u ^{4} + 4 u - 4 u ^{2} - 4 u ^{3}}{1 - u ^{2}} = 0

\frac{4 - u ^{4} + 4 u - 4 u ^{2} - 4 u ^{3}}{1 - u ^{2}} = 0 : u \approx - 0.77472 \dots, u \approx 0.94696 \dots

\frac{4 - u ^{4} + 4 u - 4 u ^{2} - 4 u ^{3}}{1 - u ^{2}} = 0

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

4 - u^{4} + 4 u - 4 u^{2} - 4 u^{3} = 0

Solve

4 - u^{4} + 4 u - 4 u^{2} - 4 u^{3} = 0 : u \approx - 0.77472 \dots, u \approx 0.94696 \dots

4 - u^{4} + 4 u - 4 u^{2} - 4 u^{3} = 0

Write in the standard form

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

- u^{4} - 4 u^{3} - 4 u^{2} + 4 u + 4 = 0

Find one solution for

- u^{4} - 4 u^{3} - 4 u^{2} + 4 u + 4 = 0

using Newton-Raphson:

u \approx - 0.77472 \dots

- u^{4} - 4 u^{3} - 4 u^{2} + 4 u + 4 = 0

Newton-Raphson Approximation Definition

f (u) = - u^{4} - 4 u^{3} - 4 u^{2} + 4 u + 4

Find

f^{^{'}} (u) : - 4 u^{3} - 12 u^{2} - 8 u + 4

\frac{d}{d u} (- u^{4} - 4 u^{3} - 4 u^{2} + 4 u + 4)

Apply the Sum/Difference Rule:

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

= - \frac{d}{d u} (u^{4}) - \frac{d}{d u} (4 u^{3}) - \frac{d}{d u} (4 u^{2}) + \frac{d}{d u} (4 u) + \frac{d}{d u} (4)

\frac{d}{d u} (u^{4}) = 4 u^{3}

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

Apply the Power Rule:

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

= 4 u^{4 - 1}

Simplify

= 4 u^{3}

\frac{d}{d u} (4 u^{3}) = 12 u^{2}

\frac{d}{d u} (4 u^{3})

Take the constant out:

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

= 4 \frac{d}{d u} (u^{3})

Apply the Power Rule:

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

= 4 \cdot 3 u^{3 - 1}

Simplify

= 12 u^{2}

\frac{d}{d u} (4 u^{2}) = 8 u

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

Take the constant out:

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

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

Apply the Power Rule:

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

= 4 \cdot 2 u^{2 - 1}

Simplify

= 8 u

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

\frac{d}{d u} (4 u)

Take the constant out:

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

= 4 \frac{d u}{d u}

Apply the common derivative:

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

= 4 \cdot 1

Simplify

= 4

\frac{d}{d u} (4) = 0

\frac{d}{d u} (4)

Derivative of a constant:

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

= 0

= - 4 u^{3} - 12 u^{2} - 8 u + 4 + 0

Simplify

= - 4 u^{3} - 12 u^{2} - 8 u + 4

Let

u_{0} = - 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 0.75 : Δ u_{1} = 0.25

f (u_{0}) = - (- 1)^{4} - 4 (- 1)^{3} - 4 (- 1)^{2} + 4 (- 1) + 4 = - 1

f^{^{'}} (u_{0}) = - 4 (- 1)^{3} - 12 (- 1)^{2} - 8 (- 1) + 4 = 4

u_{1} = - 0.75

Δ u_{1} = ∣ - 0.75 - (- 1) ∣ = 0.25

Δ u_{1} = 0.25

u_{2} = - 0.77452 \dots : Δ u_{2} = 0.02452 \dots

f (u_{1}) = - (- 0.75)^{4} - 4 (- 0.75)^{3} - 4 (- 0.75)^{2} + 4 (- 0.75) + 4 = 0.12109375

f^{^{'}} (u_{1}) = - 4 (- 0.75)^{3} - 12 (- 0.75)^{2} - 8 (- 0.75) + 4 = 4.9375

u_{2} = - 0.77452 \dots

Δ u_{2} = ∣ - 0.77452 \dots - (- 0.75) ∣ = 0.02452 \dots

Δ u_{2} = 0.02452 \dots

u_{3} = - 0.77472 \dots : Δ u_{3} = 0.00020 \dots

f (u_{2}) = - (- 0.77452 \dots)^{4} - 4 (- 0.77452 \dots)^{3} - 4 (- 0.77452 \dots)^{2} + 4 (- 0.77452 \dots) + 4 = 0.00099 \dots

f^{^{'}} (u_{2}) = - 4 (- 0.77452 \dots)^{3} - 12 (- 0.77452 \dots)^{2} - 8 (- 0.77452 \dots) + 4 = 4.85604 \dots

u_{3} = - 0.77472 \dots

Δ u_{3} = ∣ - 0.77472 \dots - (- 0.77452 \dots) ∣ = 0.00020 \dots

Δ u_{3} = 0.00020 \dots

u_{4} = - 0.77472 \dots : Δ u_{4} = 1.4564 E - 8

f (u_{3}) = - (- 0.77472 \dots)^{4} - 4 (- 0.77472 \dots)^{3} - 4 (- 0.77472 \dots)^{2} + 4 (- 0.77472 \dots) + 4 = 7.07135 E - 8

f^{^{'}} (u_{3}) = - 4 (- 0.77472 \dots)^{3} - 12 (- 0.77472 \dots)^{2} - 8 (- 0.77472 \dots) + 4 = 4.85535 \dots

u_{4} = - 0.77472 \dots

Δ u_{4} = ∣ - 0.77472 \dots - (- 0.77472 \dots) ∣ = 1.4564 E - 8

Δ u_{4} = 1.4564 E - 8

u \approx - 0.77472 \dots

Apply long division:

\frac{- u ^{4} - 4 u ^{3} - 4 u ^{2} + 4 u + 4}{u + 0.77472 \dots} = - u^{3} - 3.22527 \dots u^{2} - 1.50128 \dots u + 5.16309 \dots

- u^{3} - 3.22527 \dots u^{2} - 1.50128 \dots u + 5.16309 \dots \approx 0

Find one solution for

- u^{3} - 3.22527 \dots u^{2} - 1.50128 \dots u + 5.16309 \dots = 0

using Newton-Raphson:

u \approx 0.94696 \dots

- u^{3} - 3.22527 \dots u^{2} - 1.50128 \dots u + 5.16309 \dots = 0

Newton-Raphson Approximation Definition

f (u) = - u^{3} - 3.22527 \dots u^{2} - 1.50128 \dots u + 5.16309 \dots

Find

f^{^{'}} (u) : - 3 u^{2} - 6.45054 \dots u - 1.50128 \dots

\frac{d}{d u} (- u^{3} - 3.22527 \dots u^{2} - 1.50128 \dots u + 5.16309 \dots)

Apply the Sum/Difference Rule:

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

= - \frac{d}{d u} (u^{3}) - \frac{d}{d u} (3.22527 \dots u^{2}) - \frac{d}{d u} (1.50128 \dots u) + \frac{d}{d u} (5.16309 \dots)

\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} (3.22527 \dots u^{2}) = 6.45054 \dots u

\frac{d}{d u} (3.22527 \dots u^{2})

Take the constant out:

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

= 3.22527 \dots \frac{d}{d u} (u^{2})

Apply the Power Rule:

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

= 3.22527 \dots \cdot 2 u^{2 - 1}

Simplify

= 6.45054 \dots u

\frac{d}{d u} (1.50128 \dots u) = 1.50128 \dots

\frac{d}{d u} (1.50128 \dots u)

Take the constant out:

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

= 1.50128 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 1.50128 \dots \cdot 1

Simplify

= 1.50128 \dots

\frac{d}{d u} (5.16309 \dots) = 0

\frac{d}{d u} (5.16309 \dots)

Derivative of a constant:

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

= 0

= - 3 u^{2} - 6.45054 \dots u - 1.50128 \dots + 0

Simplify

= - 3 u^{2} - 6.45054 \dots u - 1.50128 \dots

Let

u_{0} = 3

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 1.84295 \dots : Δ u_{1} = 1.15704 \dots

f (u_{0}) = - 3^{3} - 3.22527 \dots \cdot 3^{2} - 1.50128 \dots \cdot 3 + 5.16309 \dots = - 55.36820 \dots

f^{^{'}} (u_{0}) = - 3 \cdot 3^{2} - 6.45054 \dots \cdot 3 - 1.50128 \dots = - 47.85291 \dots

u_{1} = 1.84295 \dots

Δ u_{1} = ∣ 1.84295 \dots - 3 ∣ = 1.15704 \dots

Δ u_{1} = 1.15704 \dots

u_{2} = 1.21451 \dots : Δ u_{2} = 0.62843 \dots

f (u_{1}) = - 1.84295 \dots^{3} - 3.22527 \dots \cdot 1.84295 \dots^{2} - 1.50128 \dots \cdot 1.84295 \dots + 5.16309 \dots = - 14.81774 \dots

f^{^{'}} (u_{1}) = - 3 \cdot 1.84295 \dots^{2} - 6.45054 \dots \cdot 1.84295 \dots - 1.50128 \dots = - 23.57870 \dots

u_{2} = 1.21451 \dots

Δ u_{2} = ∣ 1.21451 \dots - 1.84295 \dots ∣ = 0.62843 \dots

Δ u_{2} = 0.62843 \dots

u_{3} = 0.98130 \dots : Δ u_{3} = 0.23320 \dots

f (u_{2}) = - 1.21451 \dots^{3} - 3.22527 \dots \cdot 1.21451 \dots^{2} - 1.50128 \dots \cdot 1.21451 \dots + 5.16309 \dots = - 3.20910 \dots

f^{^{'}} (u_{2}) = - 3 \cdot 1.21451 \dots^{2} - 6.45054 \dots \cdot 1.21451 \dots - 1.50128 \dots = - 13.76067 \dots

u_{3} = 0.98130 \dots

Δ u_{3} = ∣ 0.98130 \dots - 1.21451 \dots ∣ = 0.23320 \dots

Δ u_{3} = 0.23320 \dots

u_{4} = 0.94764 \dots : Δ u_{4} = 0.03366 \dots

f (u_{3}) = - 0.98130 \dots^{3} - 3.22527 \dots \cdot 0.98130 \dots^{2} - 1.50128 \dots \cdot 0.98130 \dots + 5.16309 \dots = - 0.36088 \dots

f^{^{'}} (u_{3}) = - 3 \cdot 0.98130 \dots^{2} - 6.45054 \dots \cdot 0.98130 \dots - 1.50128 \dots = - 10.72010 \dots

u_{4} = 0.94764 \dots

Δ u_{4} = ∣ 0.94764 \dots - 0.98130 \dots ∣ = 0.03366 \dots

Δ u_{4} = 0.03366 \dots

u_{5} = 0.94696 \dots : Δ u_{5} = 0.00067 \dots

f (u_{4}) = - 0.94764 \dots^{3} - 3.22527 \dots \cdot 0.94764 \dots^{2} - 1.50128 \dots \cdot 0.94764 \dots + 5.16309 \dots = - 0.00695 \dots

f^{^{'}} (u_{4}) = - 3 \cdot 0.94764 \dots^{2} - 6.45054 \dots \cdot 0.94764 \dots - 1.50128 \dots = - 10.30814 \dots

u_{5} = 0.94696 \dots

Δ u_{5} = ∣ 0.94696 \dots - 0.94764 \dots ∣ = 0.00067 \dots

Δ u_{5} = 0.00067 \dots

u_{6} = 0.94696 \dots : Δ u_{6} = 2.68037 E - 7

f (u_{5}) = - 0.94696 \dots^{3} - 3.22527 \dots \cdot 0.94696 \dots^{2} - 1.50128 \dots \cdot 0.94696 \dots + 5.16309 \dots = - 2.76077 E - 6

f^{^{'}} (u_{5}) = - 3 \cdot 0.94696 \dots^{2} - 6.45054 \dots \cdot 0.94696 \dots - 1.50128 \dots = - 10.29995 \dots

u_{6} = 0.94696 \dots

Δ u_{6} = ∣ 0.94696 \dots - 0.94696 \dots ∣ = 2.68037 E - 7

Δ u_{6} = 2.68037 E - 7

u \approx 0.94696 \dots

Apply long division:

\frac{- u ^{3} - 3.22527 \dots u ^{2} - 1.50128 \dots u + 5.16309 \dots}{u - 0.94696 \dots} = - u^{2} - 4.17223 \dots u - 5.45225 \dots

- u^{2} - 4.17223 \dots u - 5.45225 \dots \approx 0

Find one solution for

- u^{2} - 4.17223 \dots u - 5.45225 \dots = 0

using Newton-Raphson:

No Solution for

u \in R

- u^{2} - 4.17223 \dots u - 5.45225 \dots = 0

Newton-Raphson Approximation Definition

f (u) = - u^{2} - 4.17223 \dots u - 5.45225 \dots

Find

f^{^{'}} (u) : - 2 u - 4.17223 \dots

\frac{d}{d u} (- u^{2} - 4.17223 \dots u - 5.45225 \dots)

Apply the Sum/Difference Rule:

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

= - \frac{d}{d u} (u^{2}) - \frac{d}{d u} (4.17223 \dots u) - \frac{d}{d u} (5.45225 \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} (4.17223 \dots u) = 4.17223 \dots

\frac{d}{d u} (4.17223 \dots u)

Take the constant out:

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

= 4.17223 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 4.17223 \dots \cdot 1

Simplify

= 4.17223 \dots

\frac{d}{d u} (5.45225 \dots) = 0

\frac{d}{d u} (5.45225 \dots)

Derivative of a constant:

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

= 0

= - 2 u - 4.17223 \dots - 0

Simplify

= - 2 u - 4.17223 \dots

Let

u_{0} = - 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 2.04961 \dots : Δ u_{1} = 1.04961 \dots

f (u_{0}) = - (- 1)^{2} - 4.17223 \dots (- 1) - 5.45225 \dots = - 2.28001 \dots

f^{^{'}} (u_{0}) = - 2 (- 1) - 4.17223 \dots = - 2.17223 \dots

u_{1} = - 2.04961 \dots

Δ u_{1} = ∣ - 2.04961 \dots - (- 1) ∣ = 1.04961 \dots

Δ u_{1} = 1.04961 \dots

u_{2} = - 17.14075 \dots : Δ u_{2} = 15.09113 \dots

f (u_{1}) = - (- 2.04961 \dots)^{2} - 4.17223 \dots (- 2.04961 \dots) - 5.45225 \dots = - 1.10169 \dots

f^{^{'}} (u_{1}) = - 2 (- 2.04961 \dots) - 4.17223 \dots = - 0.07300 \dots

u_{2} = - 17.14075 \dots

Δ u_{2} = ∣ - 17.14075 \dots - (- 2.04961 \dots) ∣ = 15.09113 \dots

Δ u_{2} = 15.09113 \dots

u_{3} = - 9.57688 \dots : Δ u_{3} = 7.56386 \dots

f (u_{2}) = - (- 17.14075 \dots)^{2} - 4.17223 \dots (- 17.14075 \dots) - 5.45225 \dots = - 227.74234 \dots

f^{^{'}} (u_{2}) = - 2 (- 17.14075 \dots) - 4.17223 \dots = 30.10926 \dots

u_{3} = - 9.57688 \dots

Δ u_{3} = ∣ - 9.57688 \dots - (- 17.14075 \dots) ∣ = 7.56386 \dots

Δ u_{3} = 7.56386 \dots

u_{4} = - 5.75805 \dots : Δ u_{4} = 3.81883 \dots

f (u_{3}) = - (- 9.57688 \dots)^{2} - 4.17223 \dots (- 9.57688 \dots) - 5.45225 \dots = - 57.21201 \dots

f^{^{'}} (u_{3}) = - 2 (- 9.57688 \dots) - 4.17223 \dots = 14.98154 \dots

u_{4} = - 5.75805 \dots

Δ u_{4} = ∣ - 5.75805 \dots - (- 9.57688 \dots) ∣ = 3.81883 \dots

Δ u_{4} = 3.81883 \dots

u_{5} = - 3.77225 \dots : Δ u_{5} = 1.98580 \dots

f (u_{4}) = - (- 5.75805 \dots)^{2} - 4.17223 \dots (- 5.75805 \dots) - 5.45225 \dots = - 14.58348 \dots

f^{^{'}} (u_{4}) = - 2 (- 5.75805 \dots) - 4.17223 \dots = 7.34387 \dots

u_{5} = - 3.77225 \dots

Δ u_{5} = ∣ - 3.77225 \dots - (- 5.75805 \dots) ∣ = 1.98580 \dots

Δ u_{5} = 1.98580 \dots

u_{6} = - 2.60288 \dots : Δ u_{6} = 1.16936 \dots

f (u_{5}) = - (- 3.77225 \dots)^{2} - 4.17223 \dots (- 3.77225 \dots) - 5.45225 \dots = - 3.94341 \dots

f^{^{'}} (u_{5}) = - 2 (- 3.77225 \dots) - 4.17223 \dots = 3.37226 \dots

u_{6} = - 2.60288 \dots

Δ u_{6} = ∣ - 2.60288 \dots - (- 3.77225 \dots) ∣ = 1.16936 \dots

Δ u_{6} = 1.16936 \dots

u_{7} = - 1.27984 \dots : Δ u_{7} = 1.32303 \dots

f (u_{6}) = - (- 2.60288 \dots)^{2} - 4.17223 \dots (- 2.60288 \dots) - 5.45225 \dots = - 1.36741 \dots

f^{^{'}} (u_{6}) = - 2 (- 2.60288 \dots) - 4.17223 \dots = 1.03354 \dots

u_{7} = - 1.27984 \dots

Δ u_{7} = ∣ - 1.27984 \dots - (- 2.60288 \dots) ∣ = 1.32303 \dots

Δ u_{7} = 1.32303 \dots

u_{8} = - 2.36536 \dots : Δ u_{8} = 1.08551 \dots

f (u_{7}) = - (- 1.27984 \dots)^{2} - 4.17223 \dots (- 1.27984 \dots) - 5.45225 \dots = - 1.75043 \dots

f^{^{'}} (u_{7}) = - 2 (- 1.27984 \dots) - 4.17223 \dots = - 1.61253 \dots

u_{8} = - 2.36536 \dots

Δ u_{8} = ∣ - 2.36536 \dots - (- 1.27984 \dots) ∣ = 1.08551 \dots

Δ u_{8} = 1.08551 \dots

u_{9} = - 0.25549 \dots : Δ u_{9} = 2.10986 \dots

f (u_{8}) = - (- 2.36536 \dots)^{2} - 4.17223 \dots (- 2.36536 \dots) - 5.45225 \dots = - 1.17834 \dots

f^{^{'}} (u_{8}) = - 2 (- 2.36536 \dots) - 4.17223 \dots = 0.55849 \dots

u_{9} = - 0.25549 \dots

Δ u_{9} = ∣ - 0.25549 \dots - (- 2.36536 \dots) ∣ = 2.10986 \dots

Δ u_{9} = 2.10986 \dots

u_{10} = - 1.47135 \dots : Δ u_{10} = 1.21585 \dots

f (u_{9}) = - (- 0.25549 \dots)^{2} - 4.17223 \dots (- 0.25549 \dots) - 5.45225 \dots = - 4.45153 \dots

f^{^{'}} (u_{9}) = - 2 (- 0.25549 \dots) - 4.17223 \dots = - 3.66124 \dots

u_{10} = - 1.47135 \dots

Δ u_{10} = ∣ - 1.47135 \dots - (- 0.25549 \dots) ∣ = 1.21585 \dots

Δ u_{10} = 1.21585 \dots

u_{11} = - 2.67367 \dots : Δ u_{11} = 1.20232 \dots

f (u_{10}) = - (- 1.47135 \dots)^{2} - 4.17223 \dots (- 1.47135 \dots) - 5.45225 \dots = - 1.47830 \dots

f^{^{'}} (u_{10}) = - 2 (- 1.47135 \dots) - 4.17223 \dots = - 1.22953 \dots

u_{11} = - 2.67367 \dots

Δ u_{11} = ∣ - 2.67367 \dots - (- 1.47135 \dots) ∣ = 1.20232 \dots

Δ u_{11} = 1.20232 \dots

u_{12} = - 1.44351 \dots : Δ u_{12} = 1.23016 \dots

f (u_{11}) = - (- 2.67367 \dots)^{2} - 4.17223 \dots (- 2.67367 \dots) - 5.45225 \dots = - 1.44558 \dots

f^{^{'}} (u_{11}) = - 2 (- 2.67367 \dots) - 4.17223 \dots = 1.17511 \dots

u_{12} = - 1.44351 \dots

Δ u_{12} = ∣ - 1.44351 \dots - (- 2.67367 \dots) ∣ = 1.23016 \dots

Δ u_{12} = 1.23016 \dots

Cannot find solution

The solutions are

u \approx - 0.77472 \dots, u \approx 0.94696 \dots

u \approx - 0.77472 \dots, u \approx 0.94696 \dots

Verify Solutions

Find undefined (singularity) points:

u = 1, u = - 1

Take the denominator(s) of

\frac{4 - u ^{4} + 4 u - 4 u ^{2} - 4 u ^{3}}{1 - u ^{2}}

and compare to zero

Solve

1 - u^{2} = 0 : u = 1, u = - 1

1 - u^{2} = 0

Move

1

to the right side

1 - u^{2} = 0

Subtract

1

from both sides

1 - u^{2} - 1 = 0 - 1

Simplify

- u^{2} = - 1

- u^{2} = - 1

Divide both sides by

- 1

- u^{2} = - 1

Divide both sides by

- 1

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

Simplify

u^{2} = 1

u^{2} = 1

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 1, u = - 1

1 = 1

1

Apply radical rule:

1 = 1

= 1

- 1 = - 1

- 1

Apply radical rule:

1 = 1

1 = 1

= - 1

u = 1, u = - 1

The following points are undefined

u = 1, u = - 1

Combine undefined points with solutions:

u \approx - 0.77472 \dots, u \approx 0.94696 \dots

Substitute back

u = sin (x)

sin (x) \approx - 0.77472 \dots, sin (x) \approx 0.94696 \dots

sin (x) \approx - 0.77472 \dots, sin (x) \approx 0.94696 \dots

sin (x) = - 0.77472 \dots : x = arcsin (- 0.77472 \dots) + 2 πn, x = π + arcsin (0.77472 \dots) + 2 πn

sin (x) = - 0.77472 \dots

Apply trig inverse properties

sin (x) = - 0.77472 \dots

General solutions for

sin (x) = - 0.77472 \dots

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

x = arcsin (- 0.77472 \dots) + 2 πn, x = π + arcsin (0.77472 \dots) + 2 πn

x = arcsin (- 0.77472 \dots) + 2 πn, x = π + arcsin (0.77472 \dots) + 2 πn

sin (x) = 0.94696 \dots : x = arcsin (0.94696 \dots) + 2 πn, x = π - arcsin (0.94696 \dots) + 2 πn

sin (x) = 0.94696 \dots

Apply trig inverse properties

sin (x) = 0.94696 \dots

General solutions for

sin (x) = 0.94696 \dots

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

x = arcsin (0.94696 \dots) + 2 πn, x = π - arcsin (0.94696 \dots) + 2 πn

x = arcsin (0.94696 \dots) + 2 πn, x = π - arcsin (0.94696 \dots) + 2 πn

Combine all the solutions

x = arcsin (- 0.77472 \dots) + 2 πn, x = π + arcsin (0.77472 \dots) + 2 πn, x = arcsin (0.94696 \dots) + 2 πn, x = π - arcsin (0.94696 \dots) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

sin (x) + 2 = tan (x)

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

Check the solution

arcsin (- 0.77472 \dots) + 2 πn :

False

arcsin (- 0.77472 \dots) + 2 πn

Plug in

n = 1

arcsin (- 0.77472 \dots) + 2 π 1

For

sin (x) + 2 = tan (x)

plug in

x = arcsin (- 0.77472 \dots) + 2 π 1

sin (arcsin (- 0.77472 \dots) + 2 π 1) + 2 = tan (arcsin (- 0.77472 \dots) + 2 π 1)

Refine

1.22527 \dots = - 1.22527 \dots

\Rightarrow False

Check the solution

π + arcsin (0.77472 \dots) + 2 πn :

True

π + arcsin (0.77472 \dots) + 2 πn

Plug in

n = 1

π + arcsin (0.77472 \dots) + 2 π 1

For

sin (x) + 2 = tan (x)

plug in

x = π + arcsin (0.77472 \dots) + 2 π 1

sin (π + arcsin (0.77472 \dots) + 2 π 1) + 2 = tan (π + arcsin (0.77472 \dots) + 2 π 1)

Refine

1.22527 \dots = 1.22527 \dots

\Rightarrow True

Check the solution

arcsin (0.94696 \dots) + 2 πn :

True

arcsin (0.94696 \dots) + 2 πn

Plug in

n = 1

arcsin (0.94696 \dots) + 2 π 1

For

sin (x) + 2 = tan (x)

plug in

x = arcsin (0.94696 \dots) + 2 π 1

sin (arcsin (0.94696 \dots) + 2 π 1) + 2 = tan (arcsin (0.94696 \dots) + 2 π 1)

Refine

2.94696 \dots = 2.94696 \dots

\Rightarrow True

Check the solution

π - arcsin (0.94696 \dots) + 2 πn :

False

π - arcsin (0.94696 \dots) + 2 πn

Plug in

n = 1

π - arcsin (0.94696 \dots) + 2 π 1

For

sin (x) + 2 = tan (x)

plug in

x = π - arcsin (0.94696 \dots) + 2 π 1

sin (π - arcsin (0.94696 \dots) + 2 π 1) + 2 = tan (π - arcsin (0.94696 \dots) + 2 π 1)

Refine

2.94696 \dots = - 2.94696 \dots

\Rightarrow False

x = π + arcsin (0.77472 \dots) + 2 πn, x = arcsin (0.94696 \dots) + 2 πn

Show solutions in decimal form

x = π + 0.88628 \dots + 2 πn, x = 1.24365 \dots + 2 πn

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solvefor y,ln(x)=arctan(y/x)solve for

y, ln (x) = arctan (\frac{y}{x})

Frequently Asked Questions (FAQ)

What is the general solution for sin(x)+2=tan(x) ?
The general solution for sin(x)+2=tan(x) is x=pi+0.88628…+2pin,x=1.24365…+2pin