All trigonometric functions are periodic, that is its shape repeats itself after certain interval of time. Learning graphing trigonometric functionsis useful in all areas of science and engineering. It is used for modelling various phenomenon such as population curves, waves, engines, acoustics, electronics, objects moving with simple harmonic motion, angular velocity, temperature, tides, rotation of earth, etc.
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Learning graphing trigonometric functions involvetransformation of basic trigonometric functions - sine, cosine, tangent, cosecant, secant and cotangent. Sine and cosine functions are periodic with period 2`Pi` and tangent and cotangent functions are periodic with period `Pi`.It is known that if f(x) is a periodic function with period T and a>0, then f(ax+b) is periodic with period T/a. Hence sin(ax+b), cos(ax) are periodic functions with period 2`Pi`/a. So in order to draw a periodic graph of period T, it is sufficient to draw its graph only in an interval of length T. Because once we draw in one such interval, it can be drawn completely by repeating it over the intervals of lengths T.
The concept of learning graphing trigonometric functions includes key terms - amplitude, period, vertical shift and horizontal shift.
Learn graphing trigonometric function y = sin x:
sinx is periodic with period 2`Pi`.So we need to draw the graph of y="sin(x)" in the interval [0,2`Pi` ].Initially we draw it in the interval [0,`(Pi)/(2)`].Here sin x is an increasing function.We can draw in interval [`(Pi)/(2)` , `Pi`] using the fact that,sin(`Pi`- x)=sin x.Finally we draw in the interval [`Pi`, 2`Pi`] using the fact that sin[`Pi`+ x] = - sin x.Which means the graph of y="sin" x in [`Pi`,2`Pi`] is the mirror image of graph y="sin" x in [0,`Pi`].
|sin x||0||1/2 = 0.5||1/ `sqrt(2)` = 0.707||`sqrt(3)`/2 =0.866||1|
Learn graphing trigonometric function y = cos x:
cos x is periodic with period 2`Pi`.So it is sufficient to draw the graph in the interval [0,2`Pi`].Using the table given below first draw the graph in the interval [0,`(Pi)/(2)` ].For this we use the fact that cos x is decreasing in this interval.Now, to draw the graph in the next interval [`(Pi)/(2)`, `Pi`] we use the relation cos(`Pi` - x) = - cos x.Here the graph is below x-axis.Now cos(`Pi` + x)= -cos x shows that graph of y = cos x between `Pi` and 2`Pi`is the mirror image in x-axis of its graph between 0 and `Pi`.
|cos x||1||`sqrt(3)` /2||1/`sqrt(2)`||1/2||0|
Learn graphing trigonometric function y = tan x:
tan x is periodic function with period `Pi`.It is sufficient to draw the graph over an interval [-`(Pi)/(2)`,`(Pi)/(2)`].Initially draw the graph over an interval [0,`(Pi)/(2)`].The graph of tan x is increasing in this interval.Also,as x`->``Pi`/ 2 tan x`->` `oo`.So the graph gets closer and closer to the line x = `(Pi)/(2)` as x`->` `(Pi)/(2)` .But it never touches the line x = `(Pi)/(2)`.Sincetan(-x) = -tan x, therefore, if (x,tan x) is any point on the curve y =tan x, then (-x, - tan x) will also be a point on it.This means that graph is symmetric in opposite quadrants.
Similarly, graphing remaining trigonometric functions-cosec x and sec x are also periodic functions with 2`Pi` while cot x is periodic with period `Pi`.
As x`->``+-` `(Pi)/(2)`., sec x `->` `oo`.So the curve tends closer to `oo` .These lines are known as asymptotes to the curve.Similarly,the lines x="0,x=`Pi`etc.are asymptotes of y = cosec x.
Asymptote is a line that a function approaches without intersecting at any time.
Example: Sketch the graphs of y="10" cos x and y = 10 cos(3x).
Solution:In both cases the amplitude is 10.The curve of y = 10 cos x repeats itself at x = 2`Pi`.The curve y="10cos(3x)" begins to repeat itself at x = 2`Pi`/ 3.
Example: Sketch the graph of y = sin(2x-1).
Solution:If graph is of form y="a" sin(bx+c), phase shift = -c/b.
So, here in y="sin(2x-1)," phase shift =-1/2.The graph shifts by 1/2 to the left since phase shift is negative.
First consider the graph of y="sin(2x).The" amplitude is 1 and period is 2`Pi`/ 2 = `Pi`.
The curve passes through (-0.5,0) on x-axis as we shifted the curve to the left by 1/2.
Wecan express x-coordinate and y-coordinate as function of time.This is called parametric form.Lissajous figures are special case of parametric equations,where x and y are of following form:
x = A sin(at+`delta`)
y = B sin(bt+`gamma`)