What is the general solution for sin^2(x)*sqrt(2)cos(x)=2 ?

The general solution for sin^2(x)*sqrt(2)cos(x)=2 is No Solution for x\in\mathbb{R}

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sin^2(x)*sqrt(2)cos(x)=2

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Subtract

2

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

- 2 + (1 - cos^{2} (x)) cos (x) 2 = 0

Solve by substitution

- 2 + (1 - cos^{2} (x)) cos (x) 2 = 0

Let:

cos (x) = u

- 2 + (1 - u^{2}) u 2 = 0

- 2 + (1 - u^{2}) u 2 = 0 : u \approx - 1.41421 \dots

- 2 + (1 - u^{2}) u 2 = 0

Expand

- 2 + (1 - u^{2}) u 2 : - 2 + 2 u - 2 u^{3}

- 2 + (1 - u^{2}) u 2

= - 2 + 2 u (1 - u^{2})

Expand

u 2 (1 - u^{2}) : 2 u - 2 u^{3}

u 2 (1 - u^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = u 2, b = 1, c = u^{2}

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

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

Simplify

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

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

1 \cdot 2 u = 2 u

1 \cdot 2 u

Multiply:

1 \cdot 2 = 2

= 2 u

2 u^{2} u = 2 u^{3}

2 u^{2} u

Apply exponent rule:

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

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

= 2 u^{2 + 1}

Add the numbers:

2 + 1 = 3

= 2 u^{3}

= 2 u - 2 u^{3}

= 2 u - 2 u^{3}

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

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

Write in the standard form

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

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

Find one solution for

- 1.41421 \dots u^{3} + 1.41421 \dots u - 2 = 0

using Newton-Raphson:

u \approx - 1.41421 \dots

- 1.41421 \dots u^{3} + 1.41421 \dots u - 2 = 0

Newton-Raphson Approximation Definition

f (u) = - 1.41421 \dots u^{3} + 1.41421 \dots u - 2

Find

f^{^{'}} (u) : - 4.24264 \dots u^{2} + 1.41421 \dots

\frac{d}{d u} (- 1.41421 \dots u^{3} + 1.41421 \dots u - 2)

Apply the Sum/Difference Rule:

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

= - \frac{d}{d u} (1.41421 \dots u^{3}) + \frac{d}{d u} (1.41421 \dots u) - \frac{d}{d u} (2)

\frac{d}{d u} (1.41421 \dots u^{3}) = 4.24264 \dots u^{2}

\frac{d}{d u} (1.41421 \dots u^{3})

Take the constant out:

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

= 1.41421 \dots \frac{d}{d u} (u^{3})

Apply the Power Rule:

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

= 1.41421 \dots \cdot 3 u^{3 - 1}

Simplify

= 4.24264 \dots u^{2}

\frac{d}{d u} (1.41421 \dots u) = 1.41421 \dots

\frac{d}{d u} (1.41421 \dots u)

Take the constant out:

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

= 1.41421 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 1.41421 \dots \cdot 1

Simplify

= 1.41421 \dots

\frac{d}{d u} (2) = 0

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

Derivative of a constant:

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

= 0

= - 4.24264 \dots u^{2} + 1.41421 \dots - 0

Simplify

= - 4.24264 \dots u^{2} + 1.41421 \dots

Let

u_{0} = 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 0.29289 \dots : Δ u_{1} = 0.70710 \dots

f (u_{0}) = - 1.41421 \dots \cdot 1^{3} + 1.41421 \dots \cdot 1 - 2 = - 2

f^{^{'}} (u_{0}) = - 4.24264 \dots \cdot 1^{2} + 1.41421 \dots = - 2.82842 \dots

u_{1} = 0.29289 \dots

Δ u_{1} = ∣ 0.29289 \dots - 1 ∣ = 0.70710 \dots

Δ u_{1} = 0.70710 \dots

u_{2} = 1.83663 \dots : Δ u_{2} = 1.54374 \dots

f (u_{1}) = - 1.41421 \dots \cdot 0.29289 \dots^{3} + 1.41421 \dots \cdot 0.29289 \dots - 2 = - 1.62132 \dots

f^{^{'}} (u_{1}) = - 4.24264 \dots \cdot 0.29289 \dots^{2} + 1.41421 \dots = 1.05025 \dots

u_{2} = 1.83663 \dots

Δ u_{2} = ∣ 1.83663 \dots - 0.29289 \dots ∣ = 1.54374 \dots

Δ u_{2} = 1.54374 \dots

u_{3} = 1.20361 \dots : Δ u_{3} = 0.63302 \dots

f (u_{2}) = - 1.41421 \dots \cdot 1.83663 \dots^{3} + 1.41421 \dots \cdot 1.83663 \dots - 2 = - 8.16422 \dots

f^{^{'}} (u_{2}) = - 4.24264 \dots \cdot 1.83663 \dots^{2} + 1.41421 \dots = - 12.89720 \dots

u_{3} = 1.20361 \dots

Δ u_{3} = ∣ 1.20361 \dots - 1.83663 \dots ∣ = 0.63302 \dots

Δ u_{3} = 0.63302 \dots

u_{4} = 0.61956 \dots : Δ u_{4} = 0.58404 \dots

f (u_{3}) = - 1.41421 \dots \cdot 1.20361 \dots^{3} + 1.41421 \dots \cdot 1.20361 \dots - 2 = - 2.76373 \dots

f^{^{'}} (u_{3}) = - 4.24264 \dots \cdot 1.20361 \dots^{2} + 1.41421 \dots = - 4.73203 \dots

u_{4} = 0.61956 \dots

Δ u_{4} = ∣ 0.61956 \dots - 1.20361 \dots ∣ = 0.58404 \dots

Δ u_{4} = 0.58404 \dots

u_{5} = - 6.19166 \dots : Δ u_{5} = 6.81123 \dots

f (u_{4}) = - 1.41421 \dots \cdot 0.61956 \dots^{3} + 1.41421 \dots \cdot 0.61956 \dots - 2 = - 1.46014 \dots

f^{^{'}} (u_{4}) = - 4.24264 \dots \cdot 0.61956 \dots^{2} + 1.41421 \dots = - 0.21437 \dots

u_{5} = - 6.19166 \dots

Δ u_{5} = ∣ - 6.19166 \dots - 0.61956 \dots ∣ = 6.81123 \dots

Δ u_{5} = 6.81123 \dots

u_{6} = - 4.17638 \dots : Δ u_{6} = 2.01527 \dots

f (u_{5}) = - 1.41421 \dots (- 6.19166 \dots)^{3} + 1.41421 \dots (- 6.19166 \dots) - 2 = 324.93342 \dots

f^{^{'}} (u_{5}) = - 4.24264 \dots (- 6.19166 \dots)^{2} + 1.41421 \dots = - 161.23488 \dots

u_{6} = - 4.17638 \dots

Δ u_{6} = ∣ - 4.17638 \dots - (- 6.19166 \dots) ∣ = 2.01527 \dots

Δ u_{6} = 2.01527 \dots

u_{7} = - 2.86605 \dots : Δ u_{7} = 1.31033 \dots

f (u_{6}) = - 1.41421 \dots (- 4.17638 \dots)^{3} + 1.41421 \dots (- 4.17638 \dots) - 2 = 95.11278 \dots

f^{^{'}} (u_{6}) = - 4.24264 \dots (- 4.17638 \dots)^{2} + 1.41421 \dots = - 72.58686 \dots

u_{7} = - 2.86605 \dots

Δ u_{7} = ∣ - 2.86605 \dots - (- 4.17638 \dots) ∣ = 1.31033 \dots

Δ u_{7} = 1.31033 \dots

u_{8} = - 2.05133 \dots : Δ u_{8} = 0.81472 \dots

f (u_{7}) = - 1.41421 \dots (- 2.86605 \dots)^{3} + 1.41421 \dots (- 2.86605 \dots) - 2 = 27.24108 \dots

f^{^{'}} (u_{7}) = - 4.24264 \dots (- 2.86605 \dots)^{2} + 1.41421 \dots = - 33.43606 \dots

u_{8} = - 2.05133 \dots

Δ u_{8} = ∣ - 2.05133 \dots - (- 2.86605 \dots) ∣ = 0.81472 \dots

Δ u_{8} = 0.81472 \dots

u_{9} = - 1.60687 \dots : Δ u_{9} = 0.44446 \dots

f (u_{8}) = - 1.41421 \dots (- 2.05133 \dots)^{3} + 1.41421 \dots (- 2.05133 \dots) - 2 = 7.30644 \dots

f^{^{'}} (u_{8}) = - 4.24264 \dots (- 2.05133 \dots)^{2} + 1.41421 \dots = - 16.43873 \dots

u_{9} = - 1.60687 \dots

Δ u_{9} = ∣ - 1.60687 \dots - (- 2.05133 \dots) ∣ = 0.44446 \dots

Δ u_{9} = 0.44446 \dots

u_{10} = - 1.43967 \dots : Δ u_{10} = 0.16719 \dots

f (u_{9}) = - 1.41421 \dots (- 1.60687 \dots)^{3} + 1.41421 \dots (- 1.60687 \dots) - 2 = 1.59511 \dots

f^{^{'}} (u_{9}) = - 4.24264 \dots (- 1.60687 \dots)^{2} + 1.41421 \dots = - 9.54043 \dots

u_{10} = - 1.43967 \dots

Δ u_{10} = ∣ - 1.43967 \dots - (- 1.60687 \dots) ∣ = 0.16719 \dots

Δ u_{10} = 0.16719 \dots

u_{11} = - 1.41474 \dots : Δ u_{11} = 0.02492 \dots

f (u_{10}) = - 1.41421 \dots (- 1.43967 \dots)^{3} + 1.41421 \dots (- 1.43967 \dots) - 2 = 0.18396 \dots

f^{^{'}} (u_{10}) = - 4.24264 \dots (- 1.43967 \dots)^{2} + 1.41421 \dots = - 7.37937 \dots

u_{11} = - 1.41474 \dots

Δ u_{11} = ∣ - 1.41474 \dots - (- 1.43967 \dots) ∣ = 0.02492 \dots

Δ u_{11} = 0.02492 \dots

u_{12} = - 1.41421 \dots : Δ u_{12} = 0.00053 \dots

f (u_{11}) = - 1.41421 \dots (- 1.41474 \dots)^{3} + 1.41421 \dots (- 1.41474 \dots) - 2 = 0.00377 \dots

f^{^{'}} (u_{11}) = - 4.24264 \dots (- 1.41474 \dots)^{2} + 1.41421 \dots = - 7.07747 \dots

u_{12} = - 1.41421 \dots

Δ u_{12} = ∣ - 1.41421 \dots - (- 1.41474 \dots) ∣ = 0.00053 \dots

Δ u_{12} = 0.00053 \dots

u_{13} = - 1.41421 \dots : Δ u_{13} = 2.41351 E - 7

f (u_{12}) = - 1.41421 \dots (- 1.41421 \dots)^{3} + 1.41421 \dots (- 1.41421 \dots) - 2 = 1.70661 E - 6

f^{^{'}} (u_{12}) = - 4.24264 \dots (- 1.41421 \dots)^{2} + 1.41421 \dots = - 7.07107 \dots

u_{13} = - 1.41421 \dots

Δ u_{13} = ∣ - 1.41421 \dots - (- 1.41421 \dots) ∣ = 2.41351 E - 7

Δ u_{13} = 2.41351 E - 7

u \approx - 1.41421 \dots

Apply long division:

\frac{- 2 u ^{3} + 2 u - 2}{u + 1.41421 \dots} = - 1.41421 \dots u^{2} + 2 u - 1.41421 \dots

- 1.41421 \dots u^{2} + 2 u - 1.41421 \dots \approx 0

Find one solution for

- 1.41421 \dots u^{2} + 2 u - 1.41421 \dots = 0

using Newton-Raphson:

No Solution for

u \in R

- 1.41421 \dots u^{2} + 2 u - 1.41421 \dots = 0

Newton-Raphson Approximation Definition

f (u) = - 1.41421 \dots u^{2} + 2 u - 1.41421 \dots

Find

f^{^{'}} (u) : - 2.82842 \dots u + 2

\frac{d}{d u} (- 1.41421 \dots u^{2} + 2 u - 1.41421 \dots)

Apply the Sum/Difference Rule:

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

= - \frac{d}{d u} (1.41421 \dots u^{2}) + \frac{d}{d u} (2 u) - \frac{d}{d u} (1.41421 \dots)

\frac{d}{d u} (1.41421 \dots u^{2}) = 2.82842 \dots u

\frac{d}{d u} (1.41421 \dots u^{2})

Take the constant out:

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

= 1.41421 \dots \frac{d}{d u} (u^{2})

Apply the Power Rule:

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

= 1.41421 \dots \cdot 2 u^{2 - 1}

Simplify

= 2.82842 \dots u

\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} (1.41421 \dots) = 0

\frac{d}{d u} (1.41421 \dots)

Derivative of a constant:

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

= 0

= - 2.82842 \dots u + 2 - 0

Simplify

= - 2.82842 \dots u + 2

Let

u_{0} = 2

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 1.16018 \dots : Δ u_{1} = 0.83981 \dots

f (u_{0}) = - 1.41421 \dots \cdot 2^{2} + 2 \cdot 2 - 1.41421 \dots = - 3.07106 \dots

f^{^{'}} (u_{0}) = - 2.82842 \dots \cdot 2 + 2 = - 3.65685 \dots

u_{1} = 1.16018 \dots

Δ u_{1} = ∣ 1.16018 \dots - 2 ∣ = 0.83981 \dots

Δ u_{1} = 0.83981 \dots

u_{2} = 0.38187 \dots : Δ u_{2} = 0.77831 \dots

f (u_{1}) = - 1.41421 \dots \cdot 1.16018 \dots^{2} + 2 \cdot 1.16018 \dots - 1.41421 \dots = - 0.99742 \dots

f^{^{'}} (u_{1}) = - 2.82842 \dots \cdot 1.16018 \dots + 2 = - 1.28150 \dots

u_{2} = 0.38187 \dots

Δ u_{2} = ∣ 0.38187 \dots - 1.16018 \dots ∣ = 0.77831 \dots

Δ u_{2} = 0.77831 \dots

u_{3} = 1.31316 \dots : Δ u_{3} = 0.93129 \dots

f (u_{2}) = - 1.41421 \dots \cdot 0.38187 \dots^{2} + 2 \cdot 0.38187 \dots - 1.41421 \dots = - 0.85669 \dots

f^{^{'}} (u_{2}) = - 2.82842 \dots \cdot 0.38187 \dots + 2 = 0.91990 \dots

u_{3} = 1.31316 \dots

Δ u_{3} = ∣ 1.31316 \dots - 0.38187 \dots ∣ = 0.93129 \dots

Δ u_{3} = 0.93129 \dots

u_{4} = 0.59763 \dots : Δ u_{4} = 0.71553 \dots

f (u_{3}) = - 1.41421 \dots \cdot 1.31316 \dots^{2} + 2 \cdot 1.31316 \dots - 1.41421 \dots = - 1.22655 \dots

f^{^{'}} (u_{3}) = - 2.82842 \dots \cdot 1.31316 \dots + 2 = - 1.71418 \dots

u_{4} = 0.59763 \dots

Δ u_{4} = ∣ 0.59763 \dots - 1.31316 \dots ∣ = 0.71553 \dots

Δ u_{4} = 0.71553 \dots

u_{5} = 2.93597 \dots : Δ u_{5} = 2.33834 \dots

f (u_{4}) = - 1.41421 \dots \cdot 0.59763 \dots^{2} + 2 \cdot 0.59763 \dots - 1.41421 \dots = - 0.72405 \dots

f^{^{'}} (u_{4}) = - 2.82842 \dots \cdot 0.59763 \dots + 2 = 0.30964 \dots

u_{5} = 2.93597 \dots

Δ u_{5} = ∣ 2.93597 \dots - 0.59763 \dots ∣ = 2.33834 \dots

Δ u_{5} = 2.33834 \dots

u_{6} = 1.70937 \dots : Δ u_{6} = 1.22659 \dots

f (u_{5}) = - 1.41421 \dots \cdot 2.93597 \dots^{2} + 2 \cdot 2.93597 \dots - 1.41421 \dots = - 7.73270 \dots

f^{^{'}} (u_{5}) = - 2.82842 \dots \cdot 2.93597 \dots + 2 = - 6.30418 \dots

u_{6} = 1.70937 \dots

Δ u_{6} = ∣ 1.70937 \dots - 2.93597 \dots ∣ = 1.22659 \dots

Δ u_{6} = 1.22659 \dots

u_{7} = 0.95880 \dots : Δ u_{7} = 0.75056 \dots

f (u_{6}) = - 1.41421 \dots \cdot 1.70937 \dots^{2} + 2 \cdot 1.70937 \dots - 1.41421 \dots = - 2.12774 \dots

f^{^{'}} (u_{6}) = - 2.82842 \dots \cdot 1.70937 \dots + 2 = - 2.83484 \dots

u_{7} = 0.95880 \dots

Δ u_{7} = ∣ 0.95880 \dots - 1.70937 \dots ∣ = 0.75056 \dots

Δ u_{7} = 0.75056 \dots

u_{8} = - 0.16028 \dots : Δ u_{8} = 1.11909 \dots

f (u_{7}) = - 1.41421 \dots \cdot 0.95880 \dots^{2} + 2 \cdot 0.95880 \dots - 1.41421 \dots = - 0.79670 \dots

f^{^{'}} (u_{7}) = - 2.82842 \dots \cdot 0.95880 \dots + 2 = - 0.71191 \dots

u_{8} = - 0.16028 \dots

Δ u_{8} = ∣ - 0.16028 \dots - 0.95880 \dots ∣ = 1.11909 \dots

Δ u_{8} = 1.11909 \dots

u_{9} = 0.56162 \dots : Δ u_{9} = 0.72191 \dots

f (u_{8}) = - 1.41421 \dots (- 0.16028 \dots)^{2} + 2 (- 0.16028 \dots) - 1.41421 \dots = - 1.77112 \dots

f^{^{'}} (u_{8}) = - 2.82842 \dots (- 0.16028 \dots) + 2 = 2.45336 \dots

u_{9} = 0.56162 \dots

Δ u_{9} = ∣ 0.56162 \dots - (- 0.16028 \dots) ∣ = 0.72191 \dots

Δ u_{9} = 0.72191 \dots

u_{10} = 2.35284 \dots : Δ u_{10} = 1.79121 \dots

f (u_{9}) = - 1.41421 \dots \cdot 0.56162 \dots^{2} + 2 \cdot 0.56162 \dots - 1.41421 \dots = - 0.73703 \dots

f^{^{'}} (u_{9}) = - 2.82842 \dots \cdot 0.56162 \dots + 2 = 0.41147 \dots

u_{10} = 2.35284 \dots

Δ u_{10} = ∣ 2.35284 \dots - 0.56162 \dots ∣ = 1.79121 \dots

Δ u_{10} = 1.79121 \dots

Cannot find solution

The solution is

u \approx - 1.41421 \dots

Substitute back

u = cos (x)

cos (x) \approx - 1.41421 \dots

cos (x) \approx - 1.41421 \dots

cos (x) = - 1.41421 \dots :

No Solution

cos (x) = - 1.41421 \dots

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)*sqrt(2)cos(x)=2 ?
The general solution for sin^2(x)*sqrt(2)cos(x)=2 is No Solution for x\in\mathbb{R}