What is the general solution for sin(θ)-cos(θ)=sqrt(3/2),θ[(3pi)/2 ,2pi] ?

The general solution for sin(θ)-cos(θ)=sqrt(3/2),θ[(3pi)/2 ,2pi] is No Solution for θ\in\mathbb{R}

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sin(θ)-cos(θ)=sqrt(3/2),θ[(3pi)/2 ,2pi]

Solution

sin (θ) - cos (θ) = \frac{3}{2}, θ [\frac{3 π}{2}, 2 π]

Solution

No Solution for θ \in R

Solution steps

sin (θ) - cos (θ) = \frac{3}{2}, θ [\frac{3 π}{2}, 2 π]

Rewrite using trig identities

sin (θ) - cos (θ)

sin (θ) - cos (θ) = 2 sin (θ - \frac{π}{4})

sin (θ) - cos (θ)

Rewrite as

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

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{1}{2}

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (\frac{π}{4}) sin (θ) - sin (\frac{π}{4}) cos (θ))

Use the Angle Sum identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= 2 sin (θ - \frac{π}{4})

= 2 sin (θ - \frac{π}{4})

2 sin (θ - \frac{π}{4}) = \frac{3}{2}

Divide both sides by

2

2 sin (θ - \frac{π}{4}) = \frac{3}{2}

Divide both sides by

2

\frac{2 sin ( θ - \frac{π}{4} )}{2} = \frac{\frac{3}{2}}{2}

Simplify

\frac{2 sin ( θ - \frac{π}{4} )}{2} = \frac{\frac{3}{2}}{2}

Simplify

\frac{2 sin ( θ - \frac{π}{4} )}{2} : sin (θ - \frac{π}{4})

\frac{2 sin ( θ - \frac{π}{4} )}{2}

Cancel the common factor:

2

= sin (θ - \frac{π}{4})

Simplify

\frac{\frac{3}{2}}{2} : \frac{3}{2}

\frac{\frac{3}{2}}{2}

\frac{3}{2} = \frac{3}{2}

\frac{3}{2}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{2}

= \frac{\frac{3}{2}}{2}

Apply the fraction rule:

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

= \frac{3}{2 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{3}{2}

sin (θ - \frac{π}{4}) = \frac{3}{2}

sin (θ - \frac{π}{4}) = \frac{3}{2}

sin (θ - \frac{π}{4}) = \frac{3}{2}

General solutions for

sin (θ - \frac{π}{4}) = \frac{3}{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}

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

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

Solve

θ - \frac{π}{4} = \frac{π}{3} + 2 πn : θ = 2 πn + \frac{7 π}{12}

θ - \frac{π}{4} = \frac{π}{3} + 2 πn

Move

\frac{π}{4}

to the right side

θ - \frac{π}{4} = \frac{π}{3} + 2 πn

Add

\frac{π}{4}

to both sides

θ - \frac{π}{4} + \frac{π}{4} = \frac{π}{3} + 2 πn + \frac{π}{4}

Simplify

θ - \frac{π}{4} + \frac{π}{4} = \frac{π}{3} + 2 πn + \frac{π}{4}

Simplify

θ - \frac{π}{4} + \frac{π}{4} : θ

θ - \frac{π}{4} + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} = 0

= θ

Simplify

\frac{π}{3} + 2 πn + \frac{π}{4} : 2 πn + \frac{7 π}{12}

\frac{π}{3} + 2 πn + \frac{π}{4}

Group like terms

= 2 πn + \frac{π}{3} + \frac{π}{4}

Least Common Multiplier of

3, 4 : 12

3, 4

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

3

4

= 3 \cdot 2 \cdot 2

Multiply the numbers:

3 \cdot 2 \cdot 2 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{3} :

multiply the denominator and numerator by

4

\frac{π}{3} = \frac{π 4}{3 \cdot 4} = \frac{π 4}{12}

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

= \frac{π 4}{12} + \frac{π 3}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 4 + π 3}{12}

Add similar elements:

4 π + 3 π = 7 π

= 2 πn + \frac{7 π}{12}

θ = 2 πn + \frac{7 π}{12}

θ = 2 πn + \frac{7 π}{12}

θ = 2 πn + \frac{7 π}{12}

Solve

θ - \frac{π}{4} = \frac{2 π}{3} + 2 πn : θ = 2 πn + \frac{11 π}{12}

θ - \frac{π}{4} = \frac{2 π}{3} + 2 πn

Move

\frac{π}{4}

to the right side

θ - \frac{π}{4} = \frac{2 π}{3} + 2 πn

Add

\frac{π}{4}

to both sides

θ - \frac{π}{4} + \frac{π}{4} = \frac{2 π}{3} + 2 πn + \frac{π}{4}

Simplify

θ - \frac{π}{4} + \frac{π}{4} = \frac{2 π}{3} + 2 πn + \frac{π}{4}

Simplify

θ - \frac{π}{4} + \frac{π}{4} : θ

θ - \frac{π}{4} + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} = 0

= θ

Simplify

\frac{2 π}{3} + 2 πn + \frac{π}{4} : 2 πn + \frac{11 π}{12}

\frac{2 π}{3} + 2 πn + \frac{π}{4}

Group like terms

= 2 πn + \frac{π}{4} + \frac{2 π}{3}

Least Common Multiplier of

4, 3 : 12

4, 3

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

4

3

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

For

\frac{2 π}{3} :

multiply the denominator and numerator by

4

\frac{2 π}{3} = \frac{2 π 4}{3 \cdot 4} = \frac{8 π}{12}

= \frac{π 3}{12} + \frac{8 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 + 8 π}{12}

Add similar elements:

3 π + 8 π = 11 π

= 2 πn + \frac{11 π}{12}

θ = 2 πn + \frac{11 π}{12}

θ = 2 πn + \frac{11 π}{12}

θ = 2 πn + \frac{11 π}{12}

θ = 2 πn + \frac{7 π}{12}, θ = 2 πn + \frac{11 π}{12}

Solutions for the range

θ [\frac{3 π}{2}, 2 π]

No Solution for θ \in R

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Frequently Asked Questions (FAQ)

What is the general solution for sin(θ)-cos(θ)=sqrt(3/2),θ[(3pi)/2 ,2pi] ?
The general solution for sin(θ)-cos(θ)=sqrt(3/2),θ[(3pi)/2 ,2pi] is No Solution for θ\in\mathbb{R}