Math Is Fun Forum

Al-Allo

Member

Registered: 2012-08-23

Posts: 324

trigonometry question

Hi, I have the following equation :

2*sin3a=sqrt(2)

sin3a=sqrt(2)/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

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: trigonometry question

hi Al-Allo

Because trig functions are periodic, there are an infinite number of solutions.  Usually a question will say what range of solutions are required or ask for the general solution.  I'll show the the latter here:

The principle value when sin X = (root 2) / 2 is 45 degrees or pi/4 in radians.

So allowing for the periodic nature of the sine function this leads to

a = (1/3) (45 + 2n pi)

=  15  or 135 or 255 or 375 or ......

Additionally, 180 - 45 will also leads to solutions of the form

a = (1/3) (135 + 2n pi)

= 45 or 165 or 285 or 405 or ........

So in summary, my method is

(i) find the principle value solution for 3a

(ii) find periodic additional values from this

(iii) divide by 3 to get values for 'a'

(iv) Now consider solutions in quadrants 2 or 3 or 4 and repeat steps (ii) and (iii) for these.

Hope that helps,

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob