Trig equations

From: Sareen Smith
Date: 14 Feb 1999

Subject: How do I solve sin(3x)=0.5 ?

How do I solve the equation sin(3x) = 0.5 ?
The answer in the back of the book gives six solutions. I can get some of them right but not all of them. I hope you can help because we have a whole exercise full of questions like that.

Maths Help suggests:

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:

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