1) The period of a trig. function
y=sin kx is 2pi/k. Then period of y=sin^2(pi.x/a) should be 2pi/(pi/a)=2a, but somewhere it is given as a. Which is correct?
2) The period of r=sin^3(theta/3) is given as 3pi. How is it worked out? Is it because after theta=0, the function becomes 0 only at theta=3pi (when it is sin pi=0)?
3) Is it possible for a curve to cross origin (i.e. function value y or r to be 0)more than once before completion of its entire length/cycle?
2a is correct
the period of sin^3(x) is the same as for sin(x). SO, sin^3(x/3) has period 2pi/(1/3) = 6pi
sure. sin(x) does just that. sin(pi)=0, but it takes 2pi to complete the period.
in polar coordinates, sin(theta/2) goes through the origin twice in its period of 4pi.
visit wolframalpha.com and play around with this stuff. It should help.
use the "plot" or "polar plot" command with the functions
For calculating complete length of curve r=sin^3(theta/3) I got right answer on integrating the arclength integrand from 0 to 3pi. How could that be?
No idea. Did you visit wolframalpha.com and look at the graph?
If you enter
arc length sin^3(x/3), x = 0 .. 3pi
it will give you the arc length, but that's only for 1/2 period.
I did plot the curve on Wolfram and got full curve across 3pi. Curve length is also shown full at 3pi but period is shown as 6pi.I can't understand this.
1) The period of a trigonometric function y = sin(kx) is indeed 2π/k. However, when dealing with composite functions such as y = sin^2(πx/a), the period can be different. Let's see why:
To find the period of y = sin^2(πx/a), we need to determine the values of x that repeat the function.
The function sin^2(πx/a) repeats when the argument inside the sine function, πx/a, increases by 2π. So we set up the equation:
πx/a + 2π = πx/a
Simplifying the equation, we get:
2π = 0
This means that the function repeats itself every x value, regardless of the value of a. Hence, the period of y = sin^2(πx/a) is simply a.
2) To find the period of r = sin^3(θ/3), we need to determine the values of θ that make the function repeat.
The function r = sin^3(θ/3) repeats when the argument inside the sine function, θ/3, increases by 2π. So we set up the equation:
θ/3 + 2π = θ/3
Simplifying the equation, we get:
2π = 0
Similar to the previous example, this means that the function repeats itself every θ value, regardless of the specific values of θ. Therefore, the period of r = sin^3(θ/3) is 3π.
3) Yes, it is indeed possible for a curve to cross the origin (i.e., the function value y or r to be 0) more than once before completing its entire length/cycle. This behavior is commonly observed in many functions, including trigonometric functions.
For example, the function y = sin(x) crosses the origin infinitely many times along its entire length of one cycle, which is 2π. Similarly, other functions like y = x^3 also cross the origin multiple times within their respective cycles.
The number of times a curve crosses the origin can vary based on the specific shape of the curve and the equation of the function. It may cross the origin several times, only once, or not at all.