period of
x(t)=10 sin(12 pi t)+4 cos(18 pi t)
sin 12πt has period 2π/12π = 1/6
cos 18πt has period 2π/18π = 1/9
So, the sum has period LCM(1/6,1/9) = 1/3
You can see this at
http://www.wolframalpha.com/input/?i=10+sin%2812+pi+t%29%2B4+cos%2818+pi+t%29+
what can I say?
period of first = 2π/(12π) = 1/6
period of 2nd = 2π/(18π) = 1/9
1/3 is a factor of both, so the
period = 1/3
verification:
http://www.wolframalpha.com/input/?i=10+sin%2812+pi+t%29%2B4+cos%2818+pi+t%29
Steve, that is really scary.
for 1/6 and 1/9 the L.C.M isn`t 1/3
To find the period of the given function x(t) = 10sin(12πt) + 4cos(18πt), we need to first recall some properties of trigonometric functions.
1. The period of sin(at) or cos(at) is given by 2π/a, where 'a' is the coefficient of the 't' inside the trigonometric function.
2. The period of the sum or difference of two trigonometric functions with different coefficients is determined by finding the least common multiple (LCM) of their coefficients.
Now, let's apply these properties to find the period of the given function.
The coefficient of 't' in sin(12πt) is 12π and the coefficient of 't' in cos(18πt) is 18π.
To find the LCM of 12 and 18, we can first find the prime factorization of both numbers:
12 = 2^2 * 3
18 = 2 * 3^2
Then, we take the highest power of each prime factor that appears in either number:
LCM(12, 18) = 2^2 * 3^2 = 36.
Therefore, the period of x(t) = 10sin(12πt) + 4cos(18πt) is 2π / 36 = π / 18.
Hence, the period of the given function is π / 18.