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

The general solution for -6sin(x)-4sin(2x)=0 is x=2pin,x=pi+2pin,x=2.41885…+2pin,x=-2.41885…+2pin

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Popular Trigonometry >

-6sin(x)-4sin(2x)=0

Solution

- 6 sin (x) - 4 sin (2 x) = 0

Solution

x = 2 πn, x = π + 2 πn, x = 2.41885 \dots + 2 πn, x = - 2.41885 \dots + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 138.59037 \dots^{\circ} + 36 0^{\circ} n, x = - 138.59037 \dots^{\circ} + 36 0^{\circ} n

Solution steps

- 6 sin (x) - 4 sin (2 x) = 0

Rewrite using trig identities

- 4 sin (2 x) - 6 sin (x)

Use the Double Angle identity:

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

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

Simplify

= - 8 sin (x) cos (x) - 6 sin (x)

- 6 sin (x) - 8 cos (x) sin (x) = 0

Factor

- 6 sin (x) - 8 cos (x) sin (x) : - 2 sin (x) (4 cos (x) + 3)

- 6 sin (x) - 8 cos (x) sin (x)

Rewrite

- 8

4 \cdot 2

Rewrite

- 6

3 \cdot 2

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

Factor out common term

- 2 sin (x)

= - 2 sin (x) (3 + 4 cos (x))

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

Solving each part separately

sin (x) = 0 or 4 cos (x) + 3 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

4 cos (x) + 3 = 0 : x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn

4 cos (x) + 3 = 0

Move

3

to the right side

4 cos (x) + 3 = 0

Subtract

3

from both sides

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

Simplify

4 cos (x) = - 3

4 cos (x) = - 3

Divide both sides by

4

4 cos (x) = - 3

Divide both sides by

4

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

Simplify

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

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

Apply trig inverse properties

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

General solutions for

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

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn

x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 2.41885 \dots + 2 πn, x = - 2.41885 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for -6sin(x)-4sin(2x)=0 ?
The general solution for -6sin(x)-4sin(2x)=0 is x=2pin,x=pi+2pin,x=2.41885…+2pin,x=-2.41885…+2pin