What is the general solution for 3sin(2x)-9sin(x)-4cos(x)+6=0 ?

The general solution for 3sin(2x)-9sin(x)-4cos(x)+6=0 is x=0.72972…+2pin,x=pi-0.72972…+2pin

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Popular >

3sin(2x)-9sin(x)-4cos(x)+6=0

Solution

3 sin (2 x) - 9 sin (x) - 4 cos (x) + 6 = 0

Solution

x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

Degrees

x = 41.81031 \dots^{\circ} + 36 0^{\circ} n, x = 138.18968 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 sin (2 x) - 9 sin (x) - 4 cos (x) + 6 = 0

Rewrite using trig identities

6 + 3 sin (2 x) - 4 cos (x) - 9 sin (x)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= 6 + 3 \cdot 2 sin (x) cos (x) - 4 cos (x) - 9 sin (x)

Simplify

= 6 + 6 sin (x) cos (x) - 4 cos (x) - 9 sin (x)

6 - 4 cos (x) - 9 sin (x) + 6 cos (x) sin (x) = 0

Factor

6 - 4 cos (x) - 9 sin (x) + 6 cos (x) sin (x) : (2 cos (x) - 3) (3 sin (x) - 2)

6 - 4 cos (x) - 9 sin (x) + 6 cos (x) sin (x)

= (- 4 cos (x) + 6) + (6 sin (x) cos (x) - 9 sin (x))

Factor out

3 sin (x)

from

6 sin (x) cos (x) - 9 sin (x) : 3 sin (x) (2 cos (x) - 3)

6 sin (x) cos (x) - 9 sin (x)

Rewrite

- 9

3 \cdot 3

Rewrite

6

2 \cdot 3

= 2 \cdot 3 sin (x) cos (x) + 3 \cdot 3 sin (x)

Factor out common term

3 sin (x)

= 3 sin (x) (2 cos (x) - 3)

Factor out

- 2

from

- 4 cos (x) + 6 : - 2 (2 cos (x) - 3)

- 4 cos (x) + 6

Rewrite

6

3 \cdot 2

Rewrite

- 4

2 \cdot 2

= 2 \cdot 2 cos (x) + 3 \cdot 2

Factor out common term

- 2

= - 2 (2 cos (x) - 3)

= 3 sin (x) (2 cos (x) - 3) - 2 (2 cos (x) - 3)

Factor out common term

2 cos (x) - 3

= (2 cos (x) - 3) (3 sin (x) - 2)

(2 cos (x) - 3) (3 sin (x) - 2) = 0

Solving each part separately

2 cos (x) - 3 = 0 or 3 sin (x) - 2 = 0

2 cos (x) - 3 = 0 :

No Solution

2 cos (x) - 3 = 0

Move

3

to the right side

2 cos (x) - 3 = 0

Add

3

to both sides

2 cos (x) - 3 + 3 = 0 + 3

Simplify

2 cos (x) = 3

2 cos (x) = 3

Divide both sides by

2

2 cos (x) = 3

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{3}{2}

Simplify

cos (x) = \frac{3}{2}

cos (x) = \frac{3}{2}

- 1 \leq cos (x) \leq 1

No Solution

3 sin (x) - 2 = 0 : x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

3 sin (x) - 2 = 0

Move

2

to the right side

3 sin (x) - 2 = 0

Add

2

to both sides

3 sin (x) - 2 + 2 = 0 + 2

Simplify

3 sin (x) = 2

3 sin (x) = 2

Divide both sides by

3

3 sin (x) = 2

Divide both sides by

3

\frac{3 sin ( x )}{3} = \frac{2}{3}

Simplify

sin (x) = \frac{2}{3}

sin (x) = \frac{2}{3}

Apply trig inverse properties

sin (x) = \frac{2}{3}

General solutions for

sin (x) = \frac{2}{3}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

Combine all the solutions

x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

Show solutions in decimal form

x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

Popular Examples

(10)/(sin(x))=(13)/(sin(72))

\frac{10}{sin ( x )} = \frac{13}{sin ( 7 2 ^{\circ} )}

sin(x)-sqrt(3)cos(x)=1,0<= x<2pi

sin (x) - 3 cos (x) = 1, 0 \leq x < 2 π

sin(α)+1=cos(α)

sin (α) + 1 = cos (α)

(sin(115))/(53)=(sin(S))/(83)

\frac{sin ( 11 5 ^{\circ} )}{53} = \frac{sin ( S )}{83}

2cos(x)-tan(x)=0

2 cos (x) - tan (x) = 0

Frequently Asked Questions (FAQ)

What is the general solution for 3sin(2x)-9sin(x)-4cos(x)+6=0 ?
The general solution for 3sin(2x)-9sin(x)-4cos(x)+6=0 is x=0.72972…+2pin,x=pi-0.72972…+2pin