how do you find the period of 2sin(2x) + 3cos(3x), and hence any 'addition of trig function' (with sin & cos) graph?
thanks!!
To find the period of a function containing more than one trig term, this is one option:
1. Find the period of each term,ie. 2sin(2x) is π and 3cos(3x) is 2π/3 (from the period formula 2π/n, sin(nx) or cos(nx))
Converting into degrees will make the lowest common denominator (LCD) calculation easier2. Convert each result to degrees,ie. 180° and 120° respectively (from the radian to degree formula 180x/π where x is the radian angle)
Step 3 and 4 is finding the lowest common denominator between the degree angles3. Decompose each degree angle into its prime parts,ie.
180 --> 18*10 --> 9*2*5*2 --> 2*2*3*3*5
120 --> 12 * 10 --> 4*3*5*2--> 2*2*2*3*5
4. Group these primes together, removing sequences that contain less of a given number.ie. 2*2 vs. 2*2*2 (choose 2*2*2 as it contains more 2's)
3 vs. 3*3 (choose 3*3 as it contains more 3's)
The end result will be the group 2*2*2*3*3*5, which equals 360°Optional: If you are required to have the degrees in radians5. Convert the degree result back into radians, (using the degree to radian formula πx/180, where x is the degree angle)ie. 360° becomes 2π radians
6. This result, 2π or 360° is your period of the function: 2sin(2x) + 3cos(3x)For some intuition, you can think of it like this:
If I count from 1 to 3 and simutanously a buddy counts 4-5. The pattern will repeat (the period) after the 6th count (2*3 from the LCD process)
1 2 3 1 2 3
14 5 4 5 4 5
4This is essentially the same concept.
Hope this helps