What is the general solution for (sin(θ)+cos^2(θ))^2=2sin(θ) ?

The general solution for (sin(θ)+cos^2(θ))^2=2sin(θ) is θ=0.80561…+2pin,θ=pi-0.80561…+2pin

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(sin(θ)+cos^2(θ))^2=2sin(θ)

Solution

(sin (θ) + cos^{2} (θ))^{2} = 2 sin (θ)

Solution

θ = 0.80561 \dots + 2 πn, θ = π - 0.80561 \dots + 2 πn

Degrees

θ = 46.15824 \dots^{\circ} + 36 0^{\circ} n, θ = 133.84175 \dots^{\circ} + 36 0^{\circ} n

Solution steps

(sin (θ) + cos^{2} (θ))^{2} = 2 sin (θ)

Subtract

2 sin (θ)

from both sides

(sin (θ) + cos^{2} (θ))^{2} - 2 sin (θ) = 0

Rewrite using trig identities

(cos^{2} (θ) + sin (θ))^{2} - 2 sin (θ)

Use the Pythagorean identity:

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

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

= (1 - sin^{2} (θ) + sin (θ))^{2} - 2 sin (θ)

Simplify

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

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

(1 - sin^{2} (θ) + sin (θ))^{2} = (1 - sin^{2} (θ) + sin (θ)) (1 - sin^{2} (θ) + sin (θ))

= (sin (θ) - sin^{2} (θ) + 1) (sin (θ) - sin^{2} (θ) + 1) - 2 sin (θ)

Expand

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

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

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

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

Simplify

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

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

Group like terms

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

Add similar elements:

- sin^{2} (θ) sin (θ) - sin^{2} (θ) sin (θ) = - 2 sin^{2} (θ) sin (θ)

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

Add similar elements:

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

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

Add similar elements:

1 \cdot sin (θ) + 1 \cdot sin (θ) = 2 sin (θ)

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

sin^{2} (θ) sin^{2} (θ) = sin^{4} (θ)

sin^{2} (θ) sin^{2} (θ)

Apply exponent rule:

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

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

= sin^{2 + 2} (θ)

Add the numbers:

2 + 2 = 4

= sin^{4} (θ)

2 sin^{2} (θ) sin (θ) = 2 sin^{3} (θ)

2 sin^{2} (θ) sin (θ)

Apply exponent rule:

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

sin^{2} (θ) sin (θ) = sin^{2 + 1} (θ)

= 2 sin^{2 + 1} (θ)

Add the numbers:

2 + 1 = 3

= 2 sin^{3} (θ)

sin (θ) sin (θ) = sin^{2} (θ)

sin (θ) sin (θ)

Apply exponent rule:

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

sin (θ) sin (θ) = sin^{1 + 1} (θ)

= sin^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= sin^{2} (θ)

1 \cdot 1 = 1

1 \cdot 1

Multiply the numbers:

1 \cdot 1 = 1

= 1

= - 2 sin^{2} (θ) + 2 sin (θ) + sin^{4} (θ) - 2 sin^{3} (θ) + sin^{2} (θ) + 1

Add similar elements:

- 2 sin^{2} (θ) + sin^{2} (θ) = - sin^{2} (θ)

= - sin^{2} (θ) + 2 sin (θ) + sin^{4} (θ) - 2 sin^{3} (θ) + 1

= - sin^{2} (θ) + 2 sin (θ) + sin^{4} (θ) - 2 sin^{3} (θ) + 1

= - sin^{2} (θ) + 2 sin (θ) + sin^{4} (θ) - 2 sin^{3} (θ) + 1 - 2 sin (θ)

Simplify

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

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

Group like terms

= - sin^{2} (θ) + 2 sin (θ) + sin^{4} (θ) - 2 sin^{3} (θ) - 2 sin (θ) + 1

Add similar elements:

2 sin (θ) - 2 sin (θ) = 0

= - sin^{2} (θ) + sin^{4} (θ) - 2 sin^{3} (θ) + 1

= - sin^{2} (θ) + sin^{4} (θ) - 2 sin^{3} (θ) + 1

= - sin^{2} (θ) + sin^{4} (θ) - 2 sin^{3} (θ) + 1

1 - sin^{2} (θ) + sin^{4} (θ) - 2 sin^{3} (θ) = 0

Solve by substitution

1 - sin^{2} (θ) + sin^{4} (θ) - 2 sin^{3} (θ) = 0

Let:

sin (θ) = u

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

1 - u^{2} + u^{4} - 2 u^{3} = 0 : u \approx 0.72125 \dots, u \approx 2.34859 \dots

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

Write in the standard form

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

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

Find one solution for

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

using Newton-Raphson:

u \approx 0.72125 \dots

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

Newton-Raphson Approximation Definition

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

Find

f^{^{'}} (u) : 4 u^{3} - 6 u^{2} - 2 u

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

Apply the Sum/Difference Rule:

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

= \frac{d}{d u} (u^{4}) - \frac{d}{d u} (2 u^{3}) - \frac{d}{d u} (u^{2}) + \frac{d}{d u} (1)

\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} (2 u^{3}) = 6 u^{2}

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

Take the constant out:

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

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

Apply the Power Rule:

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

= 2 \cdot 3 u^{3 - 1}

Simplify

= 6 u^{2}

\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} (1) = 0

\frac{d}{d u} (1)

Derivative of a constant:

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

= 0

= 4 u^{3} - 6 u^{2} - 2 u + 0

Simplify

= 4 u^{3} - 6 u^{2} - 2 u

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} - 2 \cdot 1^{3} - 1^{2} + 1 = - 1

f^{^{'}} (u_{0}) = 4 \cdot 1^{3} - 6 \cdot 1^{2} - 2 \cdot 1 = - 4

u_{1} = 0.75

Δ u_{1} = ∣ 0.75 - 1 ∣ = 0.25

Δ u_{1} = 0.25

u_{2} = 0.72181 \dots : Δ u_{2} = 0.02818 \dots

f (u_{1}) = 0.7 5^{4} - 2 \cdot 0.7 5^{3} - 0.7 5^{2} + 1 = - 0.08984375

f^{^{'}} (u_{1}) = 4 \cdot 0.7 5^{3} - 6 \cdot 0.7 5^{2} - 2 \cdot 0.75 = - 3.1875

u_{2} = 0.72181 \dots

Δ u_{2} = ∣ 0.72181 \dots - 0.75 ∣ = 0.02818 \dots

Δ u_{2} = 0.02818 \dots

u_{3} = 0.72125 \dots : Δ u_{3} = 0.00055 \dots

f (u_{2}) = 0.72181 \dots^{4} - 2 \cdot 0.72181 \dots^{3} - 0.72181 \dots^{2} + 1 = - 0.00171 \dots

f^{^{'}} (u_{2}) = 4 \cdot 0.72181 \dots^{3} - 6 \cdot 0.72181 \dots^{2} - 2 \cdot 0.72181 \dots = - 3.06541 \dots

u_{3} = 0.72125 \dots

Δ u_{3} = ∣ 0.72125 \dots - 0.72181 \dots ∣ = 0.00055 \dots

Δ u_{3} = 0.00055 \dots

u_{4} = 0.72125 \dots : Δ u_{4} = 2.24047 E - 7

f (u_{3}) = 0.72125 \dots^{4} - 2 \cdot 0.72125 \dots^{3} - 0.72125 \dots^{2} + 1 = - 6.86246 E - 7

f^{^{'}} (u_{3}) = 4 \cdot 0.72125 \dots^{3} - 6 \cdot 0.72125 \dots^{2} - 2 \cdot 0.72125 \dots = - 3.06295 \dots

u_{4} = 0.72125 \dots

Δ u_{4} = ∣ 0.72125 \dots - 0.72125 \dots ∣ = 2.24047 E - 7

Δ u_{4} = 2.24047 E - 7

u \approx 0.72125 \dots

Apply long division:

\frac{u ^{4} - 2 u ^{3} - u ^{2} + 1}{u - 0.72125 \dots} = u^{3} - 1.27874 \dots u^{2} - 1.92230 \dots u - 1.38647 \dots

u^{3} - 1.27874 \dots u^{2} - 1.92230 \dots u - 1.38647 \dots \approx 0

Find one solution for

u^{3} - 1.27874 \dots u^{2} - 1.92230 \dots u - 1.38647 \dots = 0

using Newton-Raphson:

u \approx 2.34859 \dots

u^{3} - 1.27874 \dots u^{2} - 1.92230 \dots u - 1.38647 \dots = 0

Newton-Raphson Approximation Definition

f (u) = u^{3} - 1.27874 \dots u^{2} - 1.92230 \dots u - 1.38647 \dots

Find

f^{^{'}} (u) : 3 u^{2} - 2.55748 \dots u - 1.92230 \dots

\frac{d}{d u} (u^{3} - 1.27874 \dots u^{2} - 1.92230 \dots u - 1.38647 \dots)

Apply the Sum/Difference Rule:

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

= \frac{d}{d u} (u^{3}) - \frac{d}{d u} (1.27874 \dots u^{2}) - \frac{d}{d u} (1.92230 \dots u) - \frac{d}{d u} (1.38647 \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} (1.27874 \dots u^{2}) = 2.55748 \dots u

\frac{d}{d u} (1.27874 \dots u^{2})

Take the constant out:

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

= 1.27874 \dots \frac{d}{d u} (u^{2})

Apply the Power Rule:

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

= 1.27874 \dots \cdot 2 u^{2 - 1}

Simplify

= 2.55748 \dots u

\frac{d}{d u} (1.92230 \dots u) = 1.92230 \dots

\frac{d}{d u} (1.92230 \dots u)

Take the constant out:

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

= 1.92230 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 1.92230 \dots \cdot 1

Simplify

= 1.92230 \dots

\frac{d}{d u} (1.38647 \dots) = 0

\frac{d}{d u} (1.38647 \dots)

Derivative of a constant:

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

= 0

= 3 u^{2} - 2.55748 \dots u - 1.92230 \dots - 0

Simplify

= 3 u^{2} - 2.55748 \dots u - 1.92230 \dots

Let

u_{0} = - 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 0.52054 \dots : Δ u_{1} = 0.47945 \dots

f (u_{0}) = (- 1)^{3} - 1.27874 \dots (- 1)^{2} - 1.92230 \dots (- 1) - 1.38647 \dots = - 1.74291 \dots

f^{^{'}} (u_{0}) = 3 (- 1)^{2} - 2.55748 \dots (- 1) - 1.92230 \dots = 3.63518 \dots

u_{1} = - 0.52054 \dots

Δ u_{1} = ∣ - 0.52054 \dots - (- 1) ∣ = 0.47945 \dots

Δ u_{1} = 0.47945 \dots

u_{2} = 3.41570 \dots : Δ u_{2} = 3.93625 \dots

f (u_{1}) = (- 0.52054 \dots)^{3} - 1.27874 \dots (- 0.52054 \dots)^{2} - 1.92230 \dots (- 0.52054 \dots) - 1.38647 \dots = - 0.87337 \dots

f^{^{'}} (u_{1}) = 3 (- 0.52054 \dots)^{2} - 2.55748 \dots (- 0.52054 \dots) - 1.92230 \dots = 0.22187 \dots

u_{2} = 3.41570 \dots

Δ u_{2} = ∣ 3.41570 \dots - (- 0.52054 \dots) ∣ = 3.93625 \dots

Δ u_{2} = 3.93625 \dots

u_{3} = 2.71820 \dots : Δ u_{3} = 0.69750 \dots

f (u_{2}) = 3.41570 \dots^{3} - 1.27874 \dots \cdot 3.41570 \dots^{2} - 1.92230 \dots \cdot 3.41570 \dots - 1.38647 \dots = 16.97956 \dots

f^{^{'}} (u_{2}) = 3 \cdot 3.41570 \dots^{2} - 2.55748 \dots \cdot 3.41570 \dots - 1.92230 \dots = 24.34322 \dots

u_{3} = 2.71820 \dots

Δ u_{3} = ∣ 2.71820 \dots - 3.41570 \dots ∣ = 0.69750 \dots

Δ u_{3} = 0.69750 \dots

u_{4} = 2.41546 \dots : Δ u_{4} = 0.30273 \dots

f (u_{3}) = 2.71820 \dots^{3} - 1.27874 \dots \cdot 2.71820 \dots^{2} - 1.92230 \dots \cdot 2.71820 \dots - 1.38647 \dots = 4.02390 \dots

f^{^{'}} (u_{3}) = 3 \cdot 2.71820 \dots^{2} - 2.55748 \dots \cdot 2.71820 \dots - 1.92230 \dots = 13.29176 \dots

u_{4} = 2.41546 \dots

Δ u_{4} = ∣ 2.41546 \dots - 2.71820 \dots ∣ = 0.30273 \dots

Δ u_{4} = 0.30273 \dots

u_{5} = 2.35139 \dots : Δ u_{5} = 0.06406 \dots

f (u_{4}) = 2.41546 \dots^{3} - 1.27874 \dots \cdot 2.41546 \dots^{2} - 1.92230 \dots \cdot 2.41546 \dots - 1.38647 \dots = 0.60242 \dots

f^{^{'}} (u_{4}) = 3 \cdot 2.41546 \dots^{2} - 2.55748 \dots \cdot 2.41546 \dots - 1.92230 \dots = 9.40356 \dots

u_{5} = 2.35139 \dots

Δ u_{5} = ∣ 2.35139 \dots - 2.41546 \dots ∣ = 0.06406 \dots

Δ u_{5} = 0.06406 \dots

u_{6} = 2.34859 \dots : Δ u_{6} = 0.00280 \dots

f (u_{5}) = 2.35139 \dots^{3} - 1.27874 \dots \cdot 2.35139 \dots^{2} - 1.92230 \dots \cdot 2.35139 \dots - 1.38647 \dots = 0.02422 \dots

f^{^{'}} (u_{5}) = 3 \cdot 2.35139 \dots^{2} - 2.55748 \dots \cdot 2.35139 \dots - 1.92230 \dots = 8.65126 \dots

u_{6} = 2.34859 \dots

Δ u_{6} = ∣ 2.34859 \dots - 2.35139 \dots ∣ = 0.00280 \dots

Δ u_{6} = 0.00280 \dots

u_{7} = 2.34859 \dots : Δ u_{7} = 5.2533 E - 6

f (u_{6}) = 2.34859 \dots^{3} - 1.27874 \dots \cdot 2.34859 \dots^{2} - 1.92230 \dots \cdot 2.34859 \dots - 1.38647 \dots = 0.00004 \dots

f^{^{'}} (u_{6}) = 3 \cdot 2.34859 \dots^{2} - 2.55748 \dots \cdot 2.34859 \dots - 1.92230 \dots = 8.61893 \dots

u_{7} = 2.34859 \dots

Δ u_{7} = ∣ 2.34859 \dots - 2.34859 \dots ∣ = 5.2533 E - 6

Δ u_{7} = 5.2533 E - 6

u_{8} = 2.34859 \dots : Δ u_{8} = 1.84659 E - 11

f (u_{7}) = 2.34859 \dots^{3} - 1.27874 \dots \cdot 2.34859 \dots^{2} - 1.92230 \dots \cdot 2.34859 \dots - 1.38647 \dots = 1.59155 E - 10

f^{^{'}} (u_{7}) = 3 \cdot 2.34859 \dots^{2} - 2.55748 \dots \cdot 2.34859 \dots - 1.92230 \dots = 8.61887 \dots

u_{8} = 2.34859 \dots

Δ u_{8} = ∣ 2.34859 \dots - 2.34859 \dots ∣ = 1.84659 E - 11

Δ u_{8} = 1.84659 E - 11

u \approx 2.34859 \dots

Apply long division:

\frac{u ^{3} - 1.27874 \dots u ^{2} - 1.92230 \dots u - 1.38647 \dots}{u - 2.34859 \dots} = u^{2} + 1.06984 \dots u + 0.59034 \dots

u^{2} + 1.06984 \dots u + 0.59034 \dots \approx 0

Find one solution for

u^{2} + 1.06984 \dots u + 0.59034 \dots = 0

using Newton-Raphson:

No Solution for

u \in R

u^{2} + 1.06984 \dots u + 0.59034 \dots = 0

Newton-Raphson Approximation Definition

f (u) = u^{2} + 1.06984 \dots u + 0.59034 \dots

Find

f^{^{'}} (u) : 2 u + 1.06984 \dots

\frac{d}{d u} (u^{2} + 1.06984 \dots u + 0.59034 \dots)

Apply the Sum/Difference Rule:

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

= \frac{d}{d u} (u^{2}) + \frac{d}{d u} (1.06984 \dots u) + \frac{d}{d u} (0.59034 \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} (1.06984 \dots u) = 1.06984 \dots

\frac{d}{d u} (1.06984 \dots u)

Take the constant out:

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

= 1.06984 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 1.06984 \dots \cdot 1

Simplify

= 1.06984 \dots

\frac{d}{d u} (0.59034 \dots) = 0

\frac{d}{d u} (0.59034 \dots)

Derivative of a constant:

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

= 0

= 2 u + 1.06984 \dots + 0

Simplify

= 2 u + 1.06984 \dots

Let

u_{0} = - 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 0.44042 \dots : Δ u_{1} = 0.55957 \dots

f (u_{0}) = (- 1)^{2} + 1.06984 \dots (- 1) + 0.59034 \dots = 0.52049 \dots

f^{^{'}} (u_{0}) = 2 (- 1) + 1.06984 \dots = - 0.93015 \dots

u_{1} = - 0.44042 \dots

Δ u_{1} = ∣ - 0.44042 \dots - (- 1) ∣ = 0.55957 \dots

Δ u_{1} = 0.55957 \dots

u_{2} = - 2.09713 \dots : Δ u_{2} = 1.65671 \dots

f (u_{1}) = (- 0.44042 \dots)^{2} + 1.06984 \dots (- 0.44042 \dots) + 0.59034 \dots = 0.31312 \dots

f^{^{'}} (u_{1}) = 2 (- 0.44042 \dots) + 1.06984 \dots = 0.18900 \dots

u_{2} = - 2.09713 \dots

Δ u_{2} = ∣ - 2.09713 \dots - (- 0.44042 \dots) ∣ = 1.65671 \dots

Δ u_{2} = 1.65671 \dots

u_{3} = - 1.21867 \dots : Δ u_{3} = 0.87846 \dots

f (u_{2}) = (- 2.09713 \dots)^{2} + 1.06984 \dots (- 2.09713 \dots) + 0.59034 \dots = 2.74470 \dots

f^{^{'}} (u_{2}) = 2 (- 2.09713 \dots) + 1.06984 \dots = - 3.12442 \dots

u_{3} = - 1.21867 \dots

Δ u_{3} = ∣ - 1.21867 \dots - (- 2.09713 \dots) ∣ = 0.87846 \dots

Δ u_{3} = 0.87846 \dots

u_{4} = - 0.65434 \dots : Δ u_{4} = 0.56432 \dots

f (u_{3}) = (- 1.21867 \dots)^{2} + 1.06984 \dots (- 1.21867 \dots) + 0.59034 \dots = 0.77170 \dots

f^{^{'}} (u_{3}) = 2 (- 1.21867 \dots) + 1.06984 \dots = - 1.36749 \dots

u_{4} = - 0.65434 \dots

Δ u_{4} = ∣ - 0.65434 \dots - (- 1.21867 \dots) ∣ = 0.56432 \dots

Δ u_{4} = 0.56432 \dots

u_{5} = 0.67895 \dots : Δ u_{5} = 1.33330 \dots

f (u_{4}) = (- 0.65434 \dots)^{2} + 1.06984 \dots (- 0.65434 \dots) + 0.59034 \dots = 0.31845 \dots

f^{^{'}} (u_{4}) = 2 (- 0.65434 \dots) + 1.06984 \dots = - 0.23884 \dots

u_{5} = 0.67895 \dots

Δ u_{5} = ∣ 0.67895 \dots - (- 0.65434 \dots) ∣ = 1.33330 \dots

Δ u_{5} = 1.33330 \dots

u_{6} = - 0.05328 \dots : Δ u_{6} = 0.73223 \dots

f (u_{5}) = 0.67895 \dots^{2} + 1.06984 \dots \cdot 0.67895 \dots + 0.59034 \dots = 1.77768 \dots

f^{^{'}} (u_{5}) = 2 \cdot 0.67895 \dots + 1.06984 \dots = 2.42775 \dots

u_{6} = - 0.05328 \dots

Δ u_{6} = ∣ - 0.05328 \dots - 0.67895 \dots ∣ = 0.73223 \dots

Δ u_{6} = 0.73223 \dots

u_{7} = - 0.60989 \dots : Δ u_{7} = 0.55661 \dots

f (u_{6}) = (- 0.05328 \dots)^{2} + 1.06984 \dots (- 0.05328 \dots) + 0.59034 \dots = 0.53617 \dots

f^{^{'}} (u_{6}) = 2 (- 0.05328 \dots) + 1.06984 \dots = 0.96327 \dots

u_{7} = - 0.60989 \dots

Δ u_{7} = ∣ - 0.60989 \dots - (- 0.05328 \dots) ∣ = 0.55661 \dots

Δ u_{7} = 0.55661 \dots

u_{8} = 1.45625 \dots : Δ u_{8} = 2.06615 \dots

f (u_{7}) = (- 0.60989 \dots)^{2} + 1.06984 \dots (- 0.60989 \dots) + 0.59034 \dots = 0.30981 \dots

f^{^{'}} (u_{7}) = 2 (- 0.60989 \dots) + 1.06984 \dots = - 0.14994 \dots

u_{8} = 1.45625 \dots

Δ u_{8} = ∣ 1.45625 \dots - (- 0.60989 \dots) ∣ = 2.06615 \dots

Δ u_{8} = 2.06615 \dots

u_{9} = 0.38427 \dots : Δ u_{9} = 1.07197 \dots

f (u_{8}) = 1.45625 \dots^{2} + 1.06984 \dots \cdot 1.45625 \dots + 0.59034 \dots = 4.26898 \dots

f^{^{'}} (u_{8}) = 2 \cdot 1.45625 \dots + 1.06984 \dots = 3.98235 \dots

u_{9} = 0.38427 \dots

Δ u_{9} = ∣ 0.38427 \dots - 1.45625 \dots ∣ = 1.07197 \dots

Δ u_{9} = 1.07197 \dots

u_{10} = - 0.24079 \dots : Δ u_{10} = 0.62506 \dots

f (u_{9}) = 0.38427 \dots^{2} + 1.06984 \dots \cdot 0.38427 \dots + 0.59034 \dots = 1.14913 \dots

f^{^{'}} (u_{9}) = 2 \cdot 0.38427 \dots + 1.06984 \dots = 1.83840 \dots

u_{10} = - 0.24079 \dots

Δ u_{10} = ∣ - 0.24079 \dots - 0.38427 \dots ∣ = 0.62506 \dots

Δ u_{10} = 0.62506 \dots

u_{11} = - 0.90496 \dots : Δ u_{11} = 0.66417 \dots

f (u_{10}) = (- 0.24079 \dots)^{2} + 1.06984 \dots (- 0.24079 \dots) + 0.59034 \dots = 0.39071 \dots

f^{^{'}} (u_{10}) = 2 (- 0.24079 \dots) + 1.06984 \dots = 0.58826 \dots

u_{11} = - 0.90496 \dots

Δ u_{11} = ∣ - 0.90496 \dots - (- 0.24079 \dots) ∣ = 0.66417 \dots

Δ u_{11} = 0.66417 \dots

Cannot find solution

The solutions are

u \approx 0.72125 \dots, u \approx 2.34859 \dots

Substitute back

u = sin (θ)

sin (θ) \approx 0.72125 \dots, sin (θ) \approx 2.34859 \dots

sin (θ) \approx 0.72125 \dots, sin (θ) \approx 2.34859 \dots

sin (θ) = 0.72125 \dots : θ = arcsin (0.72125 \dots) + 2 πn, θ = π - arcsin (0.72125 \dots) + 2 πn

sin (θ) = 0.72125 \dots

Apply trig inverse properties

sin (θ) = 0.72125 \dots

General solutions for

sin (θ) = 0.72125 \dots

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

θ = arcsin (0.72125 \dots) + 2 πn, θ = π - arcsin (0.72125 \dots) + 2 πn

θ = arcsin (0.72125 \dots) + 2 πn, θ = π - arcsin (0.72125 \dots) + 2 πn

sin (θ) = 2.34859 \dots :

No Solution

sin (θ) = 2.34859 \dots

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

θ = arcsin (0.72125 \dots) + 2 πn, θ = π - arcsin (0.72125 \dots) + 2 πn

Show solutions in decimal form

θ = 0.80561 \dots + 2 πn, θ = π - 0.80561 \dots + 2 πn

Popular Examples

2/3 =sin(2θ)

\frac{2}{3} = sin (2 θ)

sin(3x)-2=-3sin(3x)

sin (3 x) - 2 = - 3 sin (3 x)

sin(a)= 1/5

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

solvefor θ,cos(θ)= 2/3 solve for

θ, cos (θ) = \frac{2}{3}

solvefor x,arcsin(x)=arccos(y)solve for

x, arcsin (x) = arccos (y)

Frequently Asked Questions (FAQ)

What is the general solution for (sin(θ)+cos^2(θ))^2=2sin(θ) ?
The general solution for (sin(θ)+cos^2(θ))^2=2sin(θ) is θ=0.80561…+2pin,θ=pi-0.80561…+2pin