How do I solve the equation sin(3x) = 0.5 ? |
Trig equations like the one you mention actually have infinitely many solutions, because trig functions repeat themselves periodically. We assume that you require all the solutions in the interval 0 to 360 degrees (one full turn).
The first thing to note is that the angle is 3x. This means that if x itself lies between 0 and 360 degrees, then 3x lies between 0 and 1080 degrees (three full turns).
The technique that we recommend (although there are others) is to write 3x as A.
The equation is then: sinA = 0.5 with A between 0 and 1080 degrees.
In the first full turn, the solutions are A = 30 and 150 degrees. We hope you are happy with this bit.
In the second full turn (add 360), A = 390 and 510 degrees.
In the third full turn (add another 360), A = 750 and 870 degrees.
So overall, A = 30 , 150 , 390 , 510 , 750 , 870 degrees.
But remember A stands for 3x.
Therefore, 3x = 30 , 150 , 390 , 510 , 750 , 870 degrees.
So dividing by 3 gives the final list of solutions for x itself:
x = 10 , 50 , 130 , 170 , 250 , 290 degrees.
If you have a graphics calculator or a computer which will plot graphs,
plot the graph y = sin(3x) for x between 0 and 360 degrees, and superimpose
the graph y = 0.5. You can check the six solutions at the points of
intersection: