What is the general solution for sin^2(x)-cos^3(x)=0 ?

The general solution for sin^2(x)-cos^3(x)=0 is x=0.71532…+2pin,x=2pi-0.71532…+2pin

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sin^2(x)-cos^3(x)=0

Solution

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

Solution

x = 0.71532 \dots + 2 πn, x = 2 π - 0.71532 \dots + 2 πn

Degrees

x = 40.98531 \dots^{\circ} + 36 0^{\circ} n, x = 319.01468 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

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

Solve by substitution

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

Let:

cos (x) = u

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

1 - u^{2} - u^{3} = 0 : u \approx 0.75487 \dots

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

Write in the standard form

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

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

Find one solution for

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

using Newton-Raphson:

u \approx 0.75487 \dots

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

Newton-Raphson Approximation Definition

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

Find

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

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

Apply the Sum/Difference Rule:

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

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

\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} (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

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

Simplify

= - 3 u^{2} - 2 u

Let

u_{0} = 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 0.8 : Δ u_{1} = 0.2

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

f^{^{'}} (u_{0}) = - 3 \cdot 1^{2} - 2 \cdot 1 = - 5

u_{1} = 0.8

Δ u_{1} = ∣ 0.8 - 1 ∣ = 0.2

Δ u_{1} = 0.2

u_{2} = 0.75681 \dots : Δ u_{2} = 0.04318 \dots

f (u_{1}) = - 0. 8^{3} - 0. 8^{2} + 1 = - 0.152

f^{^{'}} (u_{1}) = - 3 \cdot 0. 8^{2} - 2 \cdot 0.8 = - 3.52

u_{2} = 0.75681 \dots

Δ u_{2} = ∣ 0.75681 \dots - 0.8 ∣ = 0.04318 \dots

Δ u_{2} = 0.04318 \dots

u_{3} = 0.75488 \dots : Δ u_{3} = 0.00193 \dots

f (u_{2}) = - 0.75681 \dots^{3} - 0.75681 \dots^{2} + 1 = - 0.00625 \dots

f^{^{'}} (u_{2}) = - 3 \cdot 0.75681 \dots^{2} - 2 \cdot 0.75681 \dots = - 3.23195 \dots

u_{3} = 0.75488 \dots

Δ u_{3} = ∣ 0.75488 \dots - 0.75681 \dots ∣ = 0.00193 \dots

Δ u_{3} = 0.00193 \dots

u_{4} = 0.75487 \dots : Δ u_{4} = 3.80818 E - 6

f (u_{3}) = - 0.75488 \dots^{3} - 0.75488 \dots^{2} + 1 = - 0.00001 \dots

f^{^{'}} (u_{3}) = - 3 \cdot 0.75488 \dots^{2} - 2 \cdot 0.75488 \dots = - 3.21930 \dots

u_{4} = 0.75487 \dots

Δ u_{4} = ∣ 0.75487 \dots - 0.75488 \dots ∣ = 3.80818 E - 6

Δ u_{4} = 3.80818 E - 6

u_{5} = 0.75487 \dots : Δ u_{5} = 1.47065 E - 11

f (u_{4}) = - 0.75487 \dots^{3} - 0.75487 \dots^{2} + 1 = - 4.73444 E - 11

f^{^{'}} (u_{4}) = - 3 \cdot 0.75487 \dots^{2} - 2 \cdot 0.75487 \dots = - 3.21927 \dots

u_{5} = 0.75487 \dots

Δ u_{5} = ∣ 0.75487 \dots - 0.75487 \dots ∣ = 1.47065 E - 11

Δ u_{5} = 1.47065 E - 11

u \approx 0.75487 \dots

Apply long division:

\frac{- u ^{3} - u ^{2} + 1}{u - 0.75487 \dots} = - u^{2} - 1.75487 \dots u - 1.32471 \dots

- u^{2} - 1.75487 \dots u - 1.32471 \dots \approx 0

Find one solution for

- u^{2} - 1.75487 \dots u - 1.32471 \dots = 0

using Newton-Raphson:

No Solution for

u \in R

- u^{2} - 1.75487 \dots u - 1.32471 \dots = 0

Newton-Raphson Approximation Definition

f (u) = - u^{2} - 1.75487 \dots u - 1.32471 \dots

Find

f^{^{'}} (u) : - 2 u - 1.75487 \dots

\frac{d}{d u} (- u^{2} - 1.75487 \dots u - 1.32471 \dots)

Apply the Sum/Difference Rule:

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

= - \frac{d}{d u} (u^{2}) - \frac{d}{d u} (1.75487 \dots u) - \frac{d}{d u} (1.32471 \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} (1.75487 \dots u) = 1.75487 \dots

\frac{d}{d u} (1.75487 \dots u)

Take the constant out:

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

= 1.75487 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 1.75487 \dots \cdot 1

Simplify

= 1.75487 \dots

\frac{d}{d u} (1.32471 \dots) = 0

\frac{d}{d u} (1.32471 \dots)

Derivative of a constant:

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

= 0

= - 2 u - 1.75487 \dots - 0

Simplify

= - 2 u - 1.75487 \dots

Let

u_{0} = - 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 1.32471 \dots : Δ u_{1} = 2.32471 \dots

f (u_{0}) = - (- 1)^{2} - 1.75487 \dots (- 1) - 1.32471 \dots = - 0.56984 \dots

f^{^{'}} (u_{0}) = - 2 (- 1) - 1.75487 \dots = 0.24512 \dots

u_{1} = 1.32471 \dots

Δ u_{1} = ∣ 1.32471 \dots - (- 1) ∣ = 2.32471 \dots

Δ u_{1} = 2.32471 \dots

u_{2} = 0.09766 \dots : Δ u_{2} = 1.22705 \dots

f (u_{1}) = - 1.32471 \dots^{2} - 1.75487 \dots \cdot 1.32471 \dots - 1.32471 \dots = - 5.40431 \dots

f^{^{'}} (u_{1}) = - 2 \cdot 1.32471 \dots - 1.75487 \dots = - 4.40431 \dots

u_{2} = 0.09766 \dots

Δ u_{2} = ∣ 0.09766 \dots - 1.32471 \dots ∣ = 1.22705 \dots

Δ u_{2} = 1.22705 \dots

u_{3} = - 0.67437 \dots : Δ u_{3} = 0.77204 \dots

f (u_{2}) = - 0.09766 \dots^{2} - 1.75487 \dots \cdot 0.09766 \dots - 1.32471 \dots = - 1.50565 \dots

f^{^{'}} (u_{2}) = - 2 \cdot 0.09766 \dots - 1.75487 \dots = - 1.95021 \dots

u_{3} = - 0.67437 \dots

Δ u_{3} = ∣ - 0.67437 \dots - 0.09766 \dots ∣ = 0.77204 \dots

Δ u_{3} = 0.77204 \dots

u_{4} = - 2.14204 \dots : Δ u_{4} = 1.46766 \dots

f (u_{3}) = - (- 0.67437 \dots)^{2} - 1.75487 \dots (- 0.67437 \dots) - 1.32471 \dots = - 0.59605 \dots

f^{^{'}} (u_{3}) = - 2 (- 0.67437 \dots) - 1.75487 \dots = - 0.40612 \dots

u_{4} = - 2.14204 \dots

Δ u_{4} = ∣ - 2.14204 \dots - (- 0.67437 \dots) ∣ = 1.46766 \dots

Δ u_{4} = 1.46766 \dots

u_{5} = - 1.29037 \dots : Δ u_{5} = 0.85166 \dots

f (u_{4}) = - (- 2.14204 \dots)^{2} - 1.75487 \dots (- 2.14204 \dots) - 1.32471 \dots = - 2.15403 \dots

f^{^{'}} (u_{4}) = - 2 (- 2.14204 \dots) - 1.75487 \dots = 2.52920 \dots

u_{5} = - 1.29037 \dots

Δ u_{5} = ∣ - 1.29037 \dots - (- 2.14204 \dots) ∣ = 0.85166 \dots

Δ u_{5} = 0.85166 \dots

u_{6} = - 0.41210 \dots : Δ u_{6} = 0.87826 \dots

f (u_{5}) = - (- 1.29037 \dots)^{2} - 1.75487 \dots (- 1.29037 \dots) - 1.32471 \dots = - 0.72533 \dots

f^{^{'}} (u_{5}) = - 2 (- 1.29037 \dots) - 1.75487 \dots = 0.82587 \dots

u_{6} = - 0.41210 \dots

Δ u_{6} = ∣ - 0.41210 \dots - (- 1.29037 \dots) ∣ = 0.87826 \dots

Δ u_{6} = 0.87826 \dots

u_{7} = - 1.24093 \dots : Δ u_{7} = 0.82882 \dots

f (u_{6}) = - (- 0.41210 \dots)^{2} - 1.75487 \dots (- 0.41210 \dots) - 1.32471 \dots = - 0.77135 \dots

f^{^{'}} (u_{6}) = - 2 (- 0.41210 \dots) - 1.75487 \dots = - 0.93065 \dots

u_{7} = - 1.24093 \dots

Δ u_{7} = ∣ - 1.24093 \dots - (- 0.41210 \dots) ∣ = 0.82882 \dots

Δ u_{7} = 0.82882 \dots

u_{8} = - 0.29600 \dots : Δ u_{8} = 0.94492 \dots

f (u_{7}) = - (- 1.24093 \dots)^{2} - 1.75487 \dots (- 1.24093 \dots) - 1.32471 \dots = - 0.68694 \dots

f^{^{'}} (u_{7}) = - 2 (- 1.24093 \dots) - 1.75487 \dots = 0.72698 \dots

u_{8} = - 0.29600 \dots

Δ u_{8} = ∣ - 0.29600 \dots - (- 1.24093 \dots) ∣ = 0.94492 \dots

Δ u_{8} = 0.94492 \dots

u_{9} = - 1.06383 \dots : Δ u_{9} = 0.76782 \dots

f (u_{8}) = - (- 0.29600 \dots)^{2} - 1.75487 \dots (- 0.29600 \dots) - 1.32471 \dots = - 0.89288 \dots

f^{^{'}} (u_{8}) = - 2 (- 0.29600 \dots) - 1.75487 \dots = - 1.16286 \dots

u_{9} = - 1.06383 \dots

Δ u_{9} = ∣ - 1.06383 \dots - (- 0.29600 \dots) ∣ = 0.76782 \dots

Δ u_{9} = 0.76782 \dots

u_{10} = 0.51763 \dots : Δ u_{10} = 1.58147 \dots

f (u_{9}) = - (- 1.06383 \dots)^{2} - 1.75487 \dots (- 1.06383 \dots) - 1.32471 \dots = - 0.58956 \dots

f^{^{'}} (u_{9}) = - 2 (- 1.06383 \dots) - 1.75487 \dots = 0.37279 \dots

u_{10} = 0.51763 \dots

Δ u_{10} = ∣ 0.51763 \dots - (- 1.06383 \dots) ∣ = 1.58147 \dots

Δ u_{10} = 1.58147 \dots

Cannot find solution

The solution is

u \approx 0.75487 \dots

Substitute back

u = cos (x)

cos (x) \approx 0.75487 \dots

cos (x) \approx 0.75487 \dots

cos (x) = 0.75487 \dots : x = arccos (0.75487 \dots) + 2 πn, x = 2 π - arccos (0.75487 \dots) + 2 πn

cos (x) = 0.75487 \dots

Apply trig inverse properties

cos (x) = 0.75487 \dots

General solutions for

cos (x) = 0.75487 \dots

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (0.75487 \dots) + 2 πn, x = 2 π - arccos (0.75487 \dots) + 2 πn

x = arccos (0.75487 \dots) + 2 πn, x = 2 π - arccos (0.75487 \dots) + 2 πn

Combine all the solutions

x = arccos (0.75487 \dots) + 2 πn, x = 2 π - arccos (0.75487 \dots) + 2 πn

Show solutions in decimal form

x = 0.71532 \dots + 2 πn, x = 2 π - 0.71532 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)-cos^3(x)=0 ?
The general solution for sin^2(x)-cos^3(x)=0 is x=0.71532…+2pin,x=2pi-0.71532…+2pin