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Popular >

sin^2(x/2)= 1/(2-(1/2 sin(x/2)))

Solution

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

Solution

x = - 2 \cdot 0.71476 \dots + 4 πn, x = 2 π + 2 \cdot 0.71476 \dots + 4 πn, x = 2 \cdot 0.90957 \dots + 4 πn, x = 2 π - 2 \cdot 0.90957 \dots + 4 πn

Degrees

x = - 81.90640 \dots^{\circ} + 72 0^{\circ} n, x = 441.90640 \dots^{\circ} + 72 0^{\circ} n, x = 104.22985 \dots^{\circ} + 72 0^{\circ} n, x = 255.77014 \dots^{\circ} + 72 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

sin (\frac{x}{2}) = u

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

u^{2} = \frac{1}{2 - \frac{1}{2} u} : u \approx - 0.65544 \dots, u \approx 0.78924 \dots, u \approx 3.86619 \dots

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

Simplify

\frac{1}{2 - \frac{1}{2} u} : \frac{2}{4 - u}

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

Join

2 - \frac{1}{2} u : \frac{4 - u}{2}

2 - \frac{1}{2} u

Multiply

\frac{1}{2} u : \frac{u}{2}

\frac{1}{2} u

Multiply fractions:

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

= \frac{1 \cdot u}{2}

Multiply:

1 \cdot u = u

= \frac{u}{2}

= 2 - \frac{u}{2}

Convert element to fraction:

2 = \frac{2 \cdot 2}{2}

= \frac{2 \cdot 2}{2} - \frac{u}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{2 \cdot 2 - u}{2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 - u}{2}

= \frac{1}{\frac{4 - u}{2}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{2}{4 - u}

u^{2} = \frac{2}{4 - u}

Multiply both sides by

4 - u

u^{2} = \frac{2}{4 - u}

Multiply both sides by

4 - u

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

Simplify

u^{2} (4 - u) = 2

u^{2} (4 - u) = 2

Solve

u^{2} (4 - u) = 2 : u \approx - 0.65544 \dots, u \approx 0.78924 \dots, u \approx 3.86619 \dots

u^{2} (4 - u) = 2

Expand

u^{2} (4 - u) : 4 u^{2} - u^{3}

u^{2} (4 - u)

Apply the distributive law:

a (b - c) = ab - a c

a = u^{2}, b = 4, c = u

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

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

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}

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

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

Move

2

to the left side

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

Subtract

2

from both sides

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

Simplify

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

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

Write in the standard form

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

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

Find one solution for

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

using Newton-Raphson:

u \approx - 0.65544 \dots

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

Newton-Raphson Approximation Definition

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

Find

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

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

Apply the Sum/Difference Rule:

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

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

\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} (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} (2) = 0

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

Derivative of a constant:

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

= 0

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

Simplify

= - 3 u^{2} + 8 u

Let

u_{0} = - 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 0.72727 \dots : Δ u_{1} = 0.27272 \dots

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

f^{^{'}} (u_{0}) = - 3 (- 1)^{2} + 8 (- 1) = - 11

u_{1} = - 0.72727 \dots

Δ u_{1} = ∣ - 0.72727 \dots - (- 1) ∣ = 0.27272 \dots

Δ u_{1} = 0.27272 \dots

u_{2} = - 0.65969 \dots : Δ u_{2} = 0.06757 \dots

f (u_{1}) = - (- 0.72727 \dots)^{3} + 4 (- 0.72727 \dots)^{2} - 2 = 0.50037 \dots

f^{^{'}} (u_{1}) = - 3 (- 0.72727 \dots)^{2} + 8 (- 0.72727 \dots) = - 7.40495 \dots

u_{2} = - 0.65969 \dots

Δ u_{2} = ∣ - 0.65969 \dots - (- 0.72727 \dots) ∣ = 0.06757 \dots

Δ u_{2} = 0.06757 \dots

u_{3} = - 0.65545 \dots : Δ u_{3} = 0.00424 \dots

f (u_{2}) = - (- 0.65969 \dots)^{3} + 4 (- 0.65969 \dots)^{2} - 2 = 0.02791 \dots

f^{^{'}} (u_{2}) = - 3 (- 0.65969 \dots)^{2} + 8 (- 0.65969 \dots) = - 6.58320 \dots

u_{3} = - 0.65545 \dots

Δ u_{3} = ∣ - 0.65545 \dots - (- 0.65969 \dots) ∣ = 0.00424 \dots

Δ u_{3} = 0.00424 \dots

u_{4} = - 0.65544 \dots : Δ u_{4} = 0.00001 \dots

f (u_{3}) = - (- 0.65545 \dots)^{3} + 4 (- 0.65545 \dots)^{2} - 2 = 0.00010 \dots

f^{^{'}} (u_{3}) = - 3 (- 0.65545 \dots)^{2} + 8 (- 0.65545 \dots) = - 6.53254 \dots

u_{4} = - 0.65544 \dots

Δ u_{4} = ∣ - 0.65544 \dots - (- 0.65545 \dots) ∣ = 0.00001 \dots

Δ u_{4} = 0.00001 \dots

u_{5} = - 0.65544 \dots : Δ u_{5} = 2.47138 E - 10

f (u_{4}) = - (- 0.65544 \dots)^{3} + 4 (- 0.65544 \dots)^{2} - 2 = 1.61439 E - 9

f^{^{'}} (u_{4}) = - 3 (- 0.65544 \dots)^{2} + 8 (- 0.65544 \dots) = - 6.53235 \dots

u_{5} = - 0.65544 \dots

Δ u_{5} = ∣ - 0.65544 \dots - (- 0.65544 \dots) ∣ = 2.47138 E - 10

Δ u_{5} = 2.47138 E - 10

u \approx - 0.65544 \dots

Apply long division:

\frac{- u ^{3} + 4 u ^{2} - 2}{u + 0.65544 \dots} = - u^{2} + 4.65544 \dots u - 3.05137 \dots

- u^{2} + 4.65544 \dots u - 3.05137 \dots \approx 0

Find one solution for

- u^{2} + 4.65544 \dots u - 3.05137 \dots = 0

using Newton-Raphson:

u \approx 0.78924 \dots

- u^{2} + 4.65544 \dots u - 3.05137 \dots = 0

Newton-Raphson Approximation Definition

f (u) = - u^{2} + 4.65544 \dots u - 3.05137 \dots

Find

f^{^{'}} (u) : - 2 u + 4.65544 \dots

\frac{d}{d u} (- u^{2} + 4.65544 \dots u - 3.05137 \dots)

Apply the Sum/Difference Rule:

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

= - \frac{d}{d u} (u^{2}) + \frac{d}{d u} (4.65544 \dots u) - \frac{d}{d u} (3.05137 \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} (4.65544 \dots u) = 4.65544 \dots

\frac{d}{d u} (4.65544 \dots u)

Take the constant out:

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

= 4.65544 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 4.65544 \dots \cdot 1

Simplify

= 4.65544 \dots

\frac{d}{d u} (3.05137 \dots) = 0

\frac{d}{d u} (3.05137 \dots)

Derivative of a constant:

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

= 0

= - 2 u + 4.65544 \dots - 0

Simplify

= - 2 u + 4.65544 \dots

Let

u_{0} = 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 0.77251 \dots : Δ u_{1} = 0.22748 \dots

f (u_{0}) = - 1^{2} + 4.65544 \dots \cdot 1 - 3.05137 \dots = 0.60406 \dots

f^{^{'}} (u_{0}) = - 2 \cdot 1 + 4.65544 \dots = 2.65544 \dots

u_{1} = 0.77251 \dots

Δ u_{1} = ∣ 0.77251 \dots - 1 ∣ = 0.22748 \dots

Δ u_{1} = 0.22748 \dots

u_{2} = 0.78915 \dots : Δ u_{2} = 0.01663 \dots

f (u_{1}) = - 0.77251 \dots^{2} + 4.65544 \dots \cdot 0.77251 \dots - 3.05137 \dots = - 0.05174 \dots

f^{^{'}} (u_{1}) = - 2 \cdot 0.77251 \dots + 4.65544 \dots = 3.11040 \dots

u_{2} = 0.78915 \dots

Δ u_{2} = ∣ 0.78915 \dots - 0.77251 \dots ∣ = 0.01663 \dots

Δ u_{2} = 0.01663 \dots

u_{3} = 0.78924 \dots : Δ u_{3} = 0.00008 \dots

f (u_{2}) = - 0.78915 \dots^{2} + 4.65544 \dots \cdot 0.78915 \dots - 3.05137 \dots = - 0.00027 \dots

f^{^{'}} (u_{2}) = - 2 \cdot 0.78915 \dots + 4.65544 \dots = 3.07713 \dots

u_{3} = 0.78924 \dots

Δ u_{3} = ∣ 0.78924 \dots - 0.78915 \dots ∣ = 0.00008 \dots

Δ u_{3} = 0.00008 \dots

u_{4} = 0.78924 \dots : Δ u_{4} = 2.62972 E - 9

f (u_{3}) = - 0.78924 \dots^{2} + 4.65544 \dots \cdot 0.78924 \dots - 3.05137 \dots = - 8.09152 E - 9

f^{^{'}} (u_{3}) = - 2 \cdot 0.78924 \dots + 4.65544 \dots = 3.07695 \dots

u_{4} = 0.78924 \dots

Δ u_{4} = ∣ 0.78924 \dots - 0.78924 \dots ∣ = 2.62972 E - 9

Δ u_{4} = 2.62972 E - 9

u \approx 0.78924 \dots

Apply long division:

\frac{- u ^{2} + 4.65544 \dots u - 3.05137 \dots}{u - 0.78924 \dots} = - u + 3.86619 \dots

- u + 3.86619 \dots \approx 0

u \approx 3.86619 \dots

The solutions are

u \approx - 0.65544 \dots, u \approx 0.78924 \dots, u \approx 3.86619 \dots

u \approx - 0.65544 \dots, u \approx 0.78924 \dots, u \approx 3.86619 \dots

Verify Solutions

Find undefined (singularity) points:

u = 4

Take the denominator(s) of

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

and compare to zero

Solve

2 - \frac{1}{2} u = 0 : u = 4

2 - \frac{1}{2} u = 0

Move

2

to the right side

2 - \frac{1}{2} u = 0

Subtract

2

from both sides

2 - \frac{1}{2} u - 2 = 0 - 2

Simplify

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

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

Multiply both sides by

- 2

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

Multiply both sides by

- 2

(- \frac{1}{2} u) (- 2) = (- 2) (- 2)

Simplify

u = 4

u = 4

The following points are undefined

u = 4

Combine undefined points with solutions:

u \approx - 0.65544 \dots, u \approx 0.78924 \dots, u \approx 3.86619 \dots

Substitute back

u = sin (\frac{x}{2})

sin (\frac{x}{2}) \approx - 0.65544 \dots, sin (\frac{x}{2}) \approx 0.78924 \dots, sin (\frac{x}{2}) \approx 3.86619 \dots

sin (\frac{x}{2}) \approx - 0.65544 \dots, sin (\frac{x}{2}) \approx 0.78924 \dots, sin (\frac{x}{2}) \approx 3.86619 \dots

sin (\frac{x}{2}) = - 0.65544 \dots : x = - 2 arcsin (0.65544 \dots) + 4 πn, x = 2 π + 2 arcsin (0.65544 \dots) + 4 πn

sin (\frac{x}{2}) = - 0.65544 \dots

Apply trig inverse properties

sin (\frac{x}{2}) = - 0.65544 \dots

General solutions for

sin (\frac{x}{2}) = - 0.65544 \dots

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

\frac{x}{2} = arcsin (- 0.65544 \dots) + 2 πn, \frac{x}{2} = π + arcsin (0.65544 \dots) + 2 πn

\frac{x}{2} = arcsin (- 0.65544 \dots) + 2 πn, \frac{x}{2} = π + arcsin (0.65544 \dots) + 2 πn

Solve

\frac{x}{2} = arcsin (- 0.65544 \dots) + 2 πn : x = - 2 arcsin (0.65544 \dots) + 4 πn

\frac{x}{2} = arcsin (- 0.65544 \dots) + 2 πn

Simplify

arcsin (- 0.65544 \dots) + 2 πn : - arcsin (0.65544 \dots) + 2 πn

arcsin (- 0.65544 \dots) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- 0.65544 \dots) = - arcsin (0.65544 \dots)

= - arcsin (0.65544 \dots) + 2 πn

\frac{x}{2} = - arcsin (0.65544 \dots) + 2 πn

Multiply both sides by

2

\frac{x}{2} = - arcsin (0.65544 \dots) + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = - 2 arcsin (0.65544 \dots) + 2 \cdot 2 πn

Simplify

x = - 2 arcsin (0.65544 \dots) + 4 πn

x = - 2 arcsin (0.65544 \dots) + 4 πn

Solve

\frac{x}{2} = π + arcsin (0.65544 \dots) + 2 πn : x = 2 π + 2 arcsin (0.65544 \dots) + 4 πn

\frac{x}{2} = π + arcsin (0.65544 \dots) + 2 πn

Multiply both sides by

2

\frac{x}{2} = π + arcsin (0.65544 \dots) + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 π + 2 arcsin (0.65544 \dots) + 2 \cdot 2 πn

Simplify

x = 2 π + 2 arcsin (0.65544 \dots) + 4 πn

x = 2 π + 2 arcsin (0.65544 \dots) + 4 πn

x = - 2 arcsin (0.65544 \dots) + 4 πn, x = 2 π + 2 arcsin (0.65544 \dots) + 4 πn

sin (\frac{x}{2}) = 0.78924 \dots : x = 2 arcsin (0.78924 \dots) + 4 πn, x = 2 π - 2 arcsin (0.78924 \dots) + 4 πn

sin (\frac{x}{2}) = 0.78924 \dots

Apply trig inverse properties

sin (\frac{x}{2}) = 0.78924 \dots

General solutions for

sin (\frac{x}{2}) = 0.78924 \dots

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

\frac{x}{2} = arcsin (0.78924 \dots) + 2 πn, \frac{x}{2} = π - arcsin (0.78924 \dots) + 2 πn

\frac{x}{2} = arcsin (0.78924 \dots) + 2 πn, \frac{x}{2} = π - arcsin (0.78924 \dots) + 2 πn

Solve

\frac{x}{2} = arcsin (0.78924 \dots) + 2 πn : x = 2 arcsin (0.78924 \dots) + 4 πn

\frac{x}{2} = arcsin (0.78924 \dots) + 2 πn

Multiply both sides by

2

\frac{x}{2} = arcsin (0.78924 \dots) + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 arcsin (0.78924 \dots) + 2 \cdot 2 πn

Simplify

x = 2 arcsin (0.78924 \dots) + 4 πn

x = 2 arcsin (0.78924 \dots) + 4 πn

Solve

\frac{x}{2} = π - arcsin (0.78924 \dots) + 2 πn : x = 2 π - 2 arcsin (0.78924 \dots) + 4 πn

\frac{x}{2} = π - arcsin (0.78924 \dots) + 2 πn

Multiply both sides by

2

\frac{x}{2} = π - arcsin (0.78924 \dots) + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 π - 2 arcsin (0.78924 \dots) + 2 \cdot 2 πn

Simplify

x = 2 π - 2 arcsin (0.78924 \dots) + 4 πn

x = 2 π - 2 arcsin (0.78924 \dots) + 4 πn

x = 2 arcsin (0.78924 \dots) + 4 πn, x = 2 π - 2 arcsin (0.78924 \dots) + 4 πn

sin (\frac{x}{2}) = 3.86619 \dots :

No Solution

sin (\frac{x}{2}) = 3.86619 \dots

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = - 2 arcsin (0.65544 \dots) + 4 πn, x = 2 π + 2 arcsin (0.65544 \dots) + 4 πn, x = 2 arcsin (0.78924 \dots) + 4 πn, x = 2 π - 2 arcsin (0.78924 \dots) + 4 πn

Show solutions in decimal form

x = - 2 \cdot 0.71476 \dots + 4 πn, x = 2 π + 2 \cdot 0.71476 \dots + 4 πn, x = 2 \cdot 0.90957 \dots + 4 πn, x = 2 π - 2 \cdot 0.90957 \dots + 4 πn

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