What is the general solution for 2sin^2(u)=1+sin(u),0<= t<2pi ?

Q: What is the general solution for 2sin^2(u)=1+sin(u),0<= t<2pi ?

The general solution for 2sin^2(u)=1+sin(u),0<= t<2pi is u= pi/2 ,u=(7pi)/6 ,u=(11pi)/6

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2sin^2(u)=1+sin(u),0<= t<2pi

Solution

2 sin^{2} (u) = 1 + sin (u), 0 \leq t < 2 π

Solution

u = \frac{π}{2}, u = \frac{7 π}{6}, u = \frac{11 π}{6}

Degrees

u = 9 0^{\circ}, u = 21 0^{\circ}, u = 33 0^{\circ}

Solution steps

2 sin^{2} (u) = 1 + sin (u), 0 \leq t < 2 π

Solve by substitution

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

Let:

sin (u) = v

2 v^{2} = 1 + v

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

2 v^{2} = 1 + v

Move

v

to the left side

2 v^{2} = 1 + v

Subtract

v

from both sides

2 v^{2} - v = 1 + v - v

Simplify

2 v^{2} - v = 1

2 v^{2} - v = 1

Move

1

to the left side

2 v^{2} - v = 1

Subtract

1

from both sides

2 v^{2} - v - 1 = 1 - 1

Simplify

2 v^{2} - v - 1 = 0

2 v^{2} - v - 1 = 0

Solve with the quadratic formula

2 v^{2} - v - 1 = 0

Quadratic Equation Formula:

For

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

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

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

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

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

Apply rule

- (- a) = a

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

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 2 \cdot 1 = 8

4 \cdot 2 \cdot 1

Multiply the numbers:

4 \cdot 2 \cdot 1 = 8

= 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

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

Separate the solutions

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

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

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

Apply rule

- (- a) = a

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

Add the numbers:

1 + 3 = 4

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= 1

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

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

Apply rule

- (- a) = a

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

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}

The solutions to the quadratic equation are:

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

Substitute back

v = sin (u)

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

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

sin (u) = 1, 0 \leq t < 2 π : u = \frac{π}{2}

sin (u) = 1, 0 \leq t < 2 π

General solutions for

sin (u) = 1

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

u = \frac{π}{2} + 2 πn

u = \frac{π}{2} + 2 πn

Solutions for the range

0 \leq t < 2 π

u = \frac{π}{2}

sin (u) = - \frac{1}{2}, 0 \leq t < 2 π : u = \frac{7 π}{6}, u = \frac{11 π}{6}

sin (u) = - \frac{1}{2}, 0 \leq t < 2 π

General solutions for

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

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

u = \frac{7 π}{6} + 2 πn, u = \frac{11 π}{6} + 2 πn

u = \frac{7 π}{6} + 2 πn, u = \frac{11 π}{6} + 2 πn

Solutions for the range

0 \leq t < 2 π

u = \frac{7 π}{6}, u = \frac{11 π}{6}

Combine all the solutions

u = \frac{π}{2}, u = \frac{7 π}{6}, u = \frac{11 π}{6}

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin^2(u)=1+sin(u),0<= t<2pi ?
The general solution for 2sin^2(u)=1+sin(u),0<= t<2pi is u= pi/2 ,u=(7pi)/6 ,u=(11pi)/6