What is the general solution for 9sin(x/2)+9cos(x)=0 ?

The general solution for 9sin(x/2)+9cos(x)=0 is x=(7pi)/3+4pin,x=(11pi)/3+4pin,x=pi+4pin

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9sin(x/2)+9cos(x)=0

Solution

9 sin (\frac{x}{2}) + 9 cos (x) = 0

Solution

x = \frac{7 π}{3} + 4 πn, x = \frac{11 π}{3} + 4 πn, x = π + 4 πn

Degrees

x = 42 0^{\circ} + 72 0^{\circ} n, x = 66 0^{\circ} + 72 0^{\circ} n, x = 18 0^{\circ} + 72 0^{\circ} n

Solution steps

9 sin (\frac{x}{2}) + 9 cos (x) = 0

Let:

u = \frac{x}{2}

9 sin (u) + 9 cos (2 u) = 0

Rewrite using trig identities

9 cos (2 u) + 9 sin (u)

Use the Double Angle identity:

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

= 9 (1 - 2 sin^{2} (u)) + 9 sin (u)

(1 - 2 sin^{2} (u)) \cdot 9 + 9 sin (u) = 0

Solve by substitution

(1 - 2 sin^{2} (u)) \cdot 9 + 9 sin (u) = 0

Let:

sin (u) = u

(1 - 2 u^{2}) \cdot 9 + 9 u = 0

(1 - 2 u^{2}) \cdot 9 + 9 u = 0 : u = - \frac{1}{2}, u = 1

(1 - 2 u^{2}) \cdot 9 + 9 u = 0

Expand

(1 - 2 u^{2}) \cdot 9 + 9 u : 9 - 18 u^{2} + 9 u

(1 - 2 u^{2}) \cdot 9 + 9 u

= 9 (1 - 2 u^{2}) + 9 u

Expand

9 (1 - 2 u^{2}) : 9 - 18 u^{2}

9 (1 - 2 u^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = 9, b = 1, c = 2 u^{2}

= 9 \cdot 1 - 9 \cdot 2 u^{2}

Simplify

9 \cdot 1 - 9 \cdot 2 u^{2} : 9 - 18 u^{2}

9 \cdot 1 - 9 \cdot 2 u^{2}

Multiply the numbers:

9 \cdot 1 = 9

= 9 - 9 \cdot 2 u^{2}

Multiply the numbers:

9 \cdot 2 = 18

= 9 - 18 u^{2}

= 9 - 18 u^{2}

= 9 - 18 u^{2} + 9 u

9 - 18 u^{2} + 9 u = 0

Write in the standard form

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

- 18 u^{2} + 9 u + 9 = 0

Solve with the quadratic formula

- 18 u^{2} + 9 u + 9 = 0

Quadratic Equation Formula:

For

a = - 18, b = 9, c = 9

u_{1, 2} = \frac{- 9 \pm 9 ^{2} - 4 ( - 18 ) \cdot 9}{2 ( - 18 )}

u_{1, 2} = \frac{- 9 \pm 9 ^{2} - 4 ( - 18 ) \cdot 9}{2 ( - 18 )}

9^{2} - 4 (- 18) \cdot 9 = 27

9^{2} - 4 (- 18) \cdot 9

Apply rule

- (- a) = a

= 9^{2} + 4 \cdot 18 \cdot 9

Multiply the numbers:

4 \cdot 18 \cdot 9 = 648

= 9^{2} + 648

9^{2} = 81

= 81 + 648

Add the numbers:

81 + 648 = 729

= 729

Factor the number:

729 = 2 7^{2}

= 2 7^{2}

Apply radical rule:

2 7^{2} = 27

= 27

u_{1, 2} = \frac{- 9 \pm 27}{2 ( - 18 )}

Separate the solutions

u_{1} = \frac{- 9 + 27}{2 ( - 18 )}, u_{2} = \frac{- 9 - 27}{2 ( - 18 )}

u = \frac{- 9 + 27}{2 ( - 18 )} : - \frac{1}{2}

\frac{- 9 + 27}{2 ( - 18 )}

Remove parentheses:

(- a) = - a

= \frac{- 9 + 27}{- 2 \cdot 18}

Add/Subtract the numbers:

- 9 + 27 = 18

= \frac{18}{- 2 \cdot 18}

Multiply the numbers:

2 \cdot 18 = 36

= \frac{18}{- 36}

Apply the fraction rule:

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

= - \frac{18}{36}

Cancel the common factor:

18

= - \frac{1}{2}

u = \frac{- 9 - 27}{2 ( - 18 )} : 1

\frac{- 9 - 27}{2 ( - 18 )}

Remove parentheses:

(- a) = - a

= \frac{- 9 - 27}{- 2 \cdot 18}

Subtract the numbers:

- 9 - 27 = - 36

= \frac{- 36}{- 2 \cdot 18}

Multiply the numbers:

2 \cdot 18 = 36

= \frac{- 36}{- 36}

Apply the fraction rule:

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

= \frac{36}{36}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

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

Substitute back

u = sin (u)

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

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

sin (u) = - \frac{1}{2} : u = \frac{7 π}{6} + 2 πn, u = \frac{11 π}{6} + 2 πn

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

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

sin (u) = 1 : u = \frac{π}{2} + 2 πn

sin (u) = 1

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

Combine all the solutions

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

Substitute back

u = \frac{x}{2}

\frac{x}{2} = \frac{7 π}{6} + 2 πn : x = \frac{7 π}{3} + 4 πn

\frac{x}{2} = \frac{7 π}{6} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{7 π}{6} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn : \frac{7 π}{3} + 4 πn

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{7 π}{6} = \frac{7 π}{3}

2 \cdot \frac{7 π}{6}

Multiply fractions:

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

= \frac{7 π 2}{6}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{14 π}{6}

Cancel the common factor:

2

= \frac{7 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

\frac{x}{2} = \frac{11 π}{6} + 2 πn : x = \frac{11 π}{3} + 4 πn

\frac{x}{2} = \frac{11 π}{6} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{11 π}{6} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn : \frac{11 π}{3} + 4 πn

2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{11 π}{6} = \frac{11 π}{3}

2 \cdot \frac{11 π}{6}

Multiply fractions:

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

= \frac{11 π 2}{6}

Multiply the numbers:

11 \cdot 2 = 22

= \frac{22 π}{6}

Cancel the common factor:

2

= \frac{11 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{11 π}{3} + 4 πn

x = \frac{11 π}{3} + 4 πn

x = \frac{11 π}{3} + 4 πn

x = \frac{11 π}{3} + 4 πn

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

\frac{x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

x = π + 4 πn

x = π + 4 πn

x = π + 4 πn

x = \frac{7 π}{3} + 4 πn, x = \frac{11 π}{3} + 4 πn, x = π + 4 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 9sin(x/2)+9cos(x)=0 ?
The general solution for 9sin(x/2)+9cos(x)=0 is x=(7pi)/3+4pin,x=(11pi)/3+4pin,x=pi+4pin