Calculus Animations,Graphics and Lecture Notes

Trigonometry

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Graphing Trigonometric Functions

I assume that the user is familiar with radian measure. If not at the bottom of the page is a brief discussion of radian measure.

The sine function is defined for a right triangle as the ratio of the opposite side of an angle to the hypoteneuse. 

What is the relationship between this definition and  the sine curve ?

In the Animation below we consider the motion of a partice on the unit circle. At each time a right triangle is formed with hypoteneuse 1. Therefore the value of sin(t) is precisely the y coordinate. Simultaneously We plot the y coordinate on a linear scale generating the basic Sine Curve.

Basic Sine Curve

We now do the same thing to generate the basic Cosine Curve. The Cosine is defined as the ratio of the adjacent side to the hypoteneuse. This time the value of the cosine is the x coordinate which is plotted on the vertical ont the linear graph

Basic Cosine Curve

An application - suppose we have a mass on a spring initially compressed. It is released and oscillates up and down. As the following animation shows if we plot its position vs time we get a cosine curve.

Trig functions are used to model any application in which there is periodic motion.

Mass on a Spring

The sines function is an odd Function i.e. sin(-t) = -sin(t).

In the animation below we see the sine curve generated simultaneously for positive and negative values of t.

sin(-t) = -sin(t)

The cosine function is an even function i.e.  cos(-t) = cos(t)

cos(-t) = cos(t) animation

Amplitude

In the next 2 animations we consider f(x) = asin(x) and f(x) = acos(x)

|a| is called the amplitude. The effect on the graphs is the max values are now |a| and the min values are - |a| . The zeroes and the periods don't change, however if a < 0 the graphs are reflected about the x-axis.

f(x) = asin(x)

f(x) = acos(x)

Period

Here we consider functions of the type f(x) = sin(bx) and f(x) = cos(bx)

The period  is the interval over which one sine curve or cosine curve is generated and is represented  by T. You can think of it as the time it takes to generate these curves.

For sin(x) and cos(x) T = 2π. The curves then repeat in every 2π interval .

For sin(bx) we get one curve for  0 < bx < 2π

It follows   0 < x < 2π/b    therefore T =  2π/b .

f(x) = sin(bx)

f(x) = cos(bx)

Phase Shift - Horizontal Translations

The next 2 animations show the graphs of f(x) = sin(bx+c) and cos(bx+c)

We generate one curve for  0 < bx + c < 2π

 Solving for x       -c/b < x < 2π/b -c/b  So our curves start at -c/b and end at  2π/b -c/b

so the period is still 2π/b .  The term -c/b is called the phase shift

f(x) = sin(bx+c)

f(x) = cos(bx+c)

Graph of f(x) = acsc(bx+c)

Recall  csc(x) = 1/sin(x), therefore to graph  acsc(bx+c) first graph f(x) = a sin(bx+c) and then the zeroes of  a sin(bx+c) become the vertical asymptotes of acsc(bx+c) and the 2 match up at the maxs and mins of asin(bx+c).

f(x) = acsc(ax+b)

Graph of f(x) = asec(bx+c)

Recall  sec(x) = 1/cos(x), therefore to graph  asec(bx+c) first graph f(x) = a cos(bx+c) and then the zeroes of  a cos(bx+c) become the vertical asymptotes of asec(bx+c) and the 2 match up at the maxs and mins of acos(bx+c).

f(x) = asec(bx+c)

The lecture notes for the graphs of tangent and cotangent are in the links below

Lecture Notes f(x) = tan(x)

Lecture Notes f(x) = cot(x)

Radian Measure

If you think about it why are there 360º in a single revolution? It is somewhat arbitrary--why not 100º ?  Actually its based on the number of days in a year.

In Calculus and applied mathematics a different measure is used which relates angle to distance and hence can be related to time. After all the true power of trig is modeling periodic motion where we talk about sin(t) or cos(t) where t is time.

Radian measure comes from the arclength formula s = r θ. In particular if we are on a circle of radius 1 (the unit circle) then we have simply s = θ.

The radian measure is equal to the distance traveled on the unit circle starting at the

pt (1,0) traveling counter clockwise. Therefore there are 2π radians in one revolution.

See Animation 1 below.

Radian Measure

As with the convention with degrees a clockwise rotation is considered negative.

In this case the radian measure is the negative of the distance traveled clockwise around the unit circle.

Negative Radian Measure

One of the advantage of radian measure is that it is a unitless real number.

From s = r Θ since s and r are distances whatever unit is being used cancels from both sides and so  Θ  is just a number without units.



The Smart Bunny-A very short story by Kurt Vonnegut Jr.

©2008-2010 Kelly Liakos