What is the general solution for sin^2(a)=(1-cos(a))/2 ?

The general solution for sin^2(a)=(1-cos(a))/2 is a=(2pi)/3+2pin,a=(4pi)/3+2pin,a=2pin

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sin^2(a)=(1-cos(a))/2

Solution

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

Solution

a = \frac{2 π}{3} + 2 πn, a = \frac{4 π}{3} + 2 πn, a = 2 πn

Degrees

a = 12 0^{\circ} + 36 0^{\circ} n, a = 24 0^{\circ} + 36 0^{\circ} n, a = 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

\frac{1 - cos ( a )}{2}

from both sides

sin^{2} (a) - \frac{1 - cos ( a )}{2} = 0

Simplify

sin^{2} (a) - \frac{1 - cos ( a )}{2} : \frac{2 sin ^{2} ( a ) - 1 + cos ( a )}{2}

sin^{2} (a) - \frac{1 - cos ( a )}{2}

Convert element to fraction:

sin^{2} (a) = \frac{sin ^{2} ( a ) 2}{2}

= \frac{sin ^{2} ( a ) \cdot 2}{2} - \frac{1 - cos ( a )}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ^{2} ( a ) \cdot 2 - ( 1 - cos ( a ) )}{2}

Expand

sin^{2} (a) \cdot 2 - (1 - cos (a)) : sin^{2} (a) \cdot 2 - 1 + cos (a)

sin^{2} (a) \cdot 2 - (1 - cos (a))

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

- (1 - cos (a)) : - 1 + cos (a)

- (1 - cos (a))

Distribute parentheses

= - (1) - (- cos (a))

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 1 + cos (a)

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

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

\frac{2 sin ^{2} ( a ) - 1 + cos ( a )}{2} = 0

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

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

Rewrite using trig identities

- 1 + cos (a) + 2 sin^{2} (a)

Use the Pythagorean identity:

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

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

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

Simplify

- 1 + cos (a) + 2 (1 - cos^{2} (a)) : cos (a) - 2 cos^{2} (a) + 1

- 1 + cos (a) + 2 (1 - cos^{2} (a))

Expand

2 (1 - cos^{2} (a)) : 2 - 2 cos^{2} (a)

2 (1 - cos^{2} (a))

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 1, c = cos^{2} (a)

= 2 \cdot 1 - 2 cos^{2} (a)

Multiply the numbers:

2 \cdot 1 = 2

= 2 - 2 cos^{2} (a)

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

Simplify

- 1 + cos (a) + 2 - 2 cos^{2} (a) : cos (a) - 2 cos^{2} (a) + 1

- 1 + cos (a) + 2 - 2 cos^{2} (a)

Group like terms

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

Add/Subtract the numbers:

- 1 + 2 = 1

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

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

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

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

Solve by substitution

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

Let:

cos (a) = u

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

1 + u - 2 u^{2} = 0 : u = - \frac{1}{2}, u = 1

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

Write in the standard form

a x^{2} + b x + c = 0

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 2, b = 1, c = 1

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 2 ) \cdot 1}{2 ( - 2 )}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 2 ) \cdot 1}{2 ( - 2 )}

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

1^{2} - 4 (- 2) \cdot 1

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 2) \cdot 1

Apply rule

- (- a) = a

= 1 + 4 \cdot 2 \cdot 1

Multiply the numbers:

4 \cdot 2 \cdot 1 = 8

= 1 + 8

Add the numbers:

1 + 8 = 9

= 9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

u_{1, 2} = \frac{- 1 \pm 3}{2 ( - 2 )}

Separate the solutions

u_{1} = \frac{- 1 + 3}{2 ( - 2 )}, u_{2} = \frac{- 1 - 3}{2 ( - 2 )}

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

\frac{- 1 + 3}{2 ( - 2 )}

Remove parentheses:

(- a) = - a

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

Add/Subtract the numbers:

- 1 + 3 = 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{- 4}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

u = \frac{- 1 - 3}{2 ( - 2 )} : 1

\frac{- 1 - 3}{2 ( - 2 )}

Remove parentheses:

(- a) = - a

= \frac{- 1 - 3}{- 2 \cdot 2}

Subtract the numbers:

- 1 - 3 = - 4

= \frac{- 4}{- 2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 4}{- 4}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

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

Substitute back

u = cos (a)

cos (a) = - \frac{1}{2}, cos (a) = 1

cos (a) = - \frac{1}{2}, cos (a) = 1

cos (a) = - \frac{1}{2} : a = \frac{2 π}{3} + 2 πn, a = \frac{4 π}{3} + 2 πn

cos (a) = - \frac{1}{2}

General solutions for

cos (a) = - \frac{1}{2}

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

a = \frac{2 π}{3} + 2 πn, a = \frac{4 π}{3} + 2 πn

a = \frac{2 π}{3} + 2 πn, a = \frac{4 π}{3} + 2 πn

cos (a) = 1 : a = 2 πn

cos (a) = 1

General solutions for

cos (a) = 1

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

a = 0 + 2 πn

a = 0 + 2 πn

Solve

a = 0 + 2 πn : a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn

Combine all the solutions

a = \frac{2 π}{3} + 2 πn, a = \frac{4 π}{3} + 2 πn, a = 2 πn

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\frac{sin ^{22} ( a )}{sin ^{2} ( a )} = 4 - 4 sin^{2} (a)

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tan^{3} (x) = 2

sin^3(x)=3sin(x)

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

Frequently Asked Questions (FAQ)

What is the general solution for sin^2(a)=(1-cos(a))/2 ?
The general solution for sin^2(a)=(1-cos(a))/2 is a=(2pi)/3+2pin,a=(4pi)/3+2pin,a=2pin