What is the general solution for 2sin^2(x/4)-3cos(x/4)=0 ?

The general solution for 2sin^2(x/4)-3cos(x/4)=0 is x=(4pi)/3+8pin,x=(20pi)/3+8pin

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2sin^2(x/4)-3cos(x/4)=0

Solution

2 sin^{2} (\frac{x}{4}) - 3 cos (\frac{x}{4}) = 0

Solution

x = \frac{4 π}{3} + 8 πn, x = \frac{20 π}{3} + 8 πn

Degrees

x = 24 0^{\circ} + 144 0^{\circ} n, x = 120 0^{\circ} + 144 0^{\circ} n

Solution steps

2 sin^{2} (\frac{x}{4}) - 3 cos (\frac{x}{4}) = 0

Rewrite using trig identities

2 sin^{2} (\frac{x}{4}) - 3 cos (\frac{x}{4})

Use the Pythagorean identity:

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

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

= 2 (1 - cos^{2} (\frac{x}{4})) - 3 cos (\frac{x}{4})

(1 - cos^{2} (\frac{x}{4})) \cdot 2 - 3 cos (\frac{x}{4}) = 0

Solve by substitution

(1 - cos^{2} (\frac{x}{4})) \cdot 2 - 3 cos (\frac{x}{4}) = 0

Let:

cos (\frac{x}{4}) = u

(1 - u^{2}) \cdot 2 - 3 u = 0

(1 - u^{2}) \cdot 2 - 3 u = 0 : u = - 2, u = \frac{1}{2}

(1 - u^{2}) \cdot 2 - 3 u = 0

Expand

(1 - u^{2}) \cdot 2 - 3 u : 2 - 2 u^{2} - 3 u

(1 - u^{2}) \cdot 2 - 3 u

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

Expand

2 (1 - u^{2}) : 2 - 2 u^{2}

2 (1 - u^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 1, c = u^{2}

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

Multiply the numbers:

2 \cdot 1 = 2

= 2 - 2 u^{2}

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

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 2, b = - 3, c = 2

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 ( - 2 ) \cdot 2}{2 ( - 2 )}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 ( - 2 ) \cdot 2}{2 ( - 2 )}

(- 3)^{2} - 4 (- 2) \cdot 2 = 5

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

(- 3)^{2} = 3^{2}

= 3^{2} + 4 \cdot 2 \cdot 2

Multiply the numbers:

4 \cdot 2 \cdot 2 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

Add the numbers:

9 + 16 = 25

= 25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

5^{2} = 5

= 5

u_{1, 2} = \frac{- ( - 3 ) \pm 5}{2 ( - 2 )}

Separate the solutions

u_{1} = \frac{- ( - 3 ) + 5}{2 ( - 2 )}, u_{2} = \frac{- ( - 3 ) - 5}{2 ( - 2 )}

u = \frac{- ( - 3 ) + 5}{2 ( - 2 )} : - 2

\frac{- ( - 3 ) + 5}{2 ( - 2 )}

Remove parentheses:

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

= \frac{3 + 5}{- 2 \cdot 2}

Add the numbers:

3 + 5 = 8

= \frac{8}{- 2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{8}{- 4}

Apply the fraction rule:

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

= - \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= - 2

u = \frac{- ( - 3 ) - 5}{2 ( - 2 )} : \frac{1}{2}

\frac{- ( - 3 ) - 5}{2 ( - 2 )}

Remove parentheses:

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

= \frac{3 - 5}{- 2 \cdot 2}

Subtract the numbers:

3 - 5 = - 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:

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

Substitute back

u = cos (\frac{x}{4})

cos (\frac{x}{4}) = - 2, cos (\frac{x}{4}) = \frac{1}{2}

cos (\frac{x}{4}) = - 2, cos (\frac{x}{4}) = \frac{1}{2}

cos (\frac{x}{4}) = - 2 :

No Solution

cos (\frac{x}{4}) = - 2

- 1 \leq cos (x) \leq 1

No Solution

cos (\frac{x}{4}) = \frac{1}{2} : x = \frac{4 π}{3} + 8 πn, x = \frac{20 π}{3} + 8 πn

cos (\frac{x}{4}) = \frac{1}{2}

General solutions for

cos (\frac{x}{4}) = \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}

\frac{x}{4} = \frac{π}{3} + 2 πn, \frac{x}{4} = \frac{5 π}{3} + 2 πn

\frac{x}{4} = \frac{π}{3} + 2 πn, \frac{x}{4} = \frac{5 π}{3} + 2 πn

Solve

\frac{x}{4} = \frac{π}{3} + 2 πn : x = \frac{4 π}{3} + 8 πn

\frac{x}{4} = \frac{π}{3} + 2 πn

Multiply both sides by

4

\frac{x}{4} = \frac{π}{3} + 2 πn

Multiply both sides by

4

\frac{4 x}{4} = 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Simplify

\frac{4 x}{4} = 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

4 \cdot \frac{π}{3} + 4 \cdot 2 πn : \frac{4 π}{3} + 8 πn

4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Multiply

4 \cdot \frac{π}{3} : \frac{4 π}{3}

4 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 4}{3}

= \frac{4 π}{3} + 4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= \frac{4 π}{3} + 8 πn

x = \frac{4 π}{3} + 8 πn

x = \frac{4 π}{3} + 8 πn

x = \frac{4 π}{3} + 8 πn

Solve

\frac{x}{4} = \frac{5 π}{3} + 2 πn : x = \frac{20 π}{3} + 8 πn

\frac{x}{4} = \frac{5 π}{3} + 2 πn

Multiply both sides by

4

\frac{x}{4} = \frac{5 π}{3} + 2 πn

Multiply both sides by

4

\frac{4 x}{4} = 4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

Simplify

\frac{4 x}{4} = 4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn : \frac{20 π}{3} + 8 πn

4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

4 \cdot \frac{5 π}{3} = \frac{20 π}{3}

4 \cdot \frac{5 π}{3}

Multiply fractions:

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

= \frac{5 π 4}{3}

Multiply the numbers:

5 \cdot 4 = 20

= \frac{20 π}{3}

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= \frac{20 π}{3} + 8 πn

x = \frac{20 π}{3} + 8 πn

x = \frac{20 π}{3} + 8 πn

x = \frac{20 π}{3} + 8 πn

x = \frac{4 π}{3} + 8 πn, x = \frac{20 π}{3} + 8 πn

Combine all the solutions

x = \frac{4 π}{3} + 8 πn, x = \frac{20 π}{3} + 8 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin^2(x/4)-3cos(x/4)=0 ?
The general solution for 2sin^2(x/4)-3cos(x/4)=0 is x=(4pi)/3+8pin,x=(20pi)/3+8pin