What is the general solution for 6sin(2t)+9sin(t)=0 ?

The general solution for 6sin(2t)+9sin(t)=0 is t=2pin,t=pi+2pin,t=2.41885…+2pin,t=-2.41885…+2pin

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Popular >

6sin(2t)+9sin(t)=0

Solution

6 sin (2 t) + 9 sin (t) = 0

Solution

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

Degrees

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

Solution steps

6 sin (2 t) + 9 sin (t) = 0

Rewrite using trig identities

6 sin (2 t) + 9 sin (t)

Use the Double Angle identity:

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

= 6 \cdot 2 sin (t) cos (t) + 9 sin (t)

Simplify

= 12 sin (t) cos (t) + 9 sin (t)

9 sin (t) + 12 cos (t) sin (t) = 0

Factor

9 sin (t) + 12 cos (t) sin (t) : 3 sin (t) (4 cos (t) + 3)

9 sin (t) + 12 cos (t) sin (t)

Rewrite

12

4 \cdot 3

Rewrite

9

3 \cdot 3

= 3 \cdot 3 sin (t) + 4 \cdot 3 sin (t) cos (t)

Factor out common term

3 sin (t)

= 3 sin (t) (3 + 4 cos (t))

3 sin (t) (4 cos (t) + 3) = 0

Solving each part separately

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

sin (t) = 0 : t = 2 πn, t = π + 2 πn

sin (t) = 0

General solutions for

sin (t) = 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}

t = 0 + 2 πn, t = π + 2 πn

t = 0 + 2 πn, t = π + 2 πn

Solve

t = 0 + 2 πn : t = 2 πn

t = 0 + 2 πn

0 + 2 πn = 2 πn

t = 2 πn

t = 2 πn, t = π + 2 πn

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

4 cos (t) + 3 = 0

Move

3

to the right side

4 cos (t) + 3 = 0

Subtract

3

from both sides

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

Simplify

4 cos (t) = - 3

4 cos (t) = - 3

Divide both sides by

4

4 cos (t) = - 3

Divide both sides by

4

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

Simplify

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

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

Apply trig inverse properties

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

General solutions for

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

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

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

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

Combine all the solutions

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

Show solutions in decimal form

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

Popular Examples

sin(2x+60)+sin(x+30)=0

sin (2 x + 60) + sin (x + 30) = 0

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

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

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

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

4sec(x)+8=0

4 sec (x) + 8 = 0

solvefor x,ln(y)=e^ysin(x)solve for

x, ln (y) = e^{y} sin (x)

Frequently Asked Questions (FAQ)

What is the general solution for 6sin(2t)+9sin(t)=0 ?
The general solution for 6sin(2t)+9sin(t)=0 is t=2pin,t=pi+2pin,t=2.41885…+2pin,t=-2.41885…+2pin