Solve Trigonometry Equation | 2sin3a=sqrt(2) | Geometrical vs Symbolic Methods

[chemistry1]

Hi, I have the following equation :

2*sin3a=sqrt(2)

I was able to solve it in a geometrical way(giving 6 solutions.) but I have no idea on how to solve it in a symbolic way...(which would take clearly less time to do ) Is there any way ?

thank you

 

[SteamKing]

If you know how to apply the arcsine, sure.

 

[chemistry1]

Could you develop please ?I know arcsine... ( btw, it's from 0 to 2pi for the solutions)

 

[SteamKing]

Could you develop please ?I know arcsine... ( btw, it's from 0 to 2pi for the solutions)

If sin(θ) = a, where -1 <= a <= 1; then

arcsin(a) = θ, where θ is the principal angle

 

[haruspex]

Do you happen to know an angle for which the sine is 1/√2?

 

[HallsofIvy]

Hi, I have the following equation :

2*sin3a=sqrt(2)

I was able to solve it in a geometrical way(giving 6 solutions.) but I have no idea on how to solve it in a symbolic way...(which would take clearly less time to do ) Is there any way ?

thank you

The point you seem to be missing is that if you divide both sides by 2 you have
$$sin(3a)= \frac{\sqrt{2}}{2}= \frac{1}{\sqrt{2}}$$
(hence haruspex's question)

Perhaps it would make more sense as "solve $sin(\theta)= \sqrt{2}/2$ for $\theta$". Of course once you have found $\theta$, solve $3a= \theta$.

(In order to find all a between 0 and $2\pi$, you may want to find all $\theta= 3a$ between 0 and $6\pi$.)

 

[CompuChip]

There is a way that always works: if you can write the right hand side ($\tfrac{1}{\sqrt{2}}$ in this case) as sin(b), then your equation is of the form

sin(3a) = sin(b)

and the general solution is

$$3a = b + 2 \pi k, \text{ or } 3a = \pi - b + 2 \pi k$$

for all $k = 0, 1, -1, 2, -2, 3, -3, \ldots$.

Similarly the general solution for cos(x) = cos(y) is

$$x = y + 2 \pi k, \text{ or } x = - y + 2 \pi k$$

(the only difference being the $\pi$ in the second branch).