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The curves

A cosine curve is plotted with vertical values relative to the top grid line. It extends from -360 degrees on the left to +360 degrees on the right.

A sine curve is plotted with vertical values relative to the bottom grid line. It also extends from -360 degrees on the left to +360 degrees on theright. (Note once again that Figure 16 was flipped horizontally to crate a mirror image.)

Return values for the Math.asin, Math.acos, and Math.atan methods

I told you earlier that the Math.asin method returns a value between -PI/2 and PI/2. However, I didn't tell you that the Math.acos method returns a value between 0 and PI, or that the Math.atan method returns a value between -PI/2 and PI/2. You now have enough informationto understand why this is true.

Smooth curves

If you examine the two curves that you have just plotted, you can surmise that the sine and cosine functions are smooth curves whose values range between-1 and +1 inclusive. For every possible value between -1 and +1, there is an angle in the range -PI/2 and PI/2 whose sine value matches that value. There isalso an angle in the range 0 and PI whose cosine value matches that value.

(Although you haven't plotted the curve for the tangent, a similar situation holds there also.)

An infinite number of angles

Therefore, given a specific numeric value between -1 and +1, there are an infinite number of angles whose sine and cosine values match thatnumeric value and the method has no way of distinguishing between them. Therefore, the Math.asin method returns the matching angle that is closest to zero and the Math.acos method returns the matching positive angle that is closest to zero.

What can we learn from this?

One important thing that we can learn is there is no difference between the sine or cosine of an angle and the sine or cosine of a different anglethat differs from the original angle by 360 degrees. Thus, the Math.asin and Math.acos methods cannot be used to distinguish between angles that differ by 360 degrees. (As you learned above, the situation involving the Math.asin and Math.acos methods is even more stringent than that.)

One-quarter cycle contains all of the information

Another thing that we can learn is that once you know the shape of the cosine curve from 0 degrees to 90 degrees, you have enough information to construct theentire cosine curve and the entire sine curve across any range of angles. Every possible value or the negative of every possible value that can occur in a sineor cosine curve occurs in the cosine curve between 0 degrees and 90 degrees. Furthermore, the order of those values is also well defined.

Think about these relationships

You should think about these kinds of relationships. As I mentioned earlier, as long as we are working with angles between 0 and 90 degrees, everything isrelatively straightforward. However, once we start working with angles between 90 degrees and 360 degrees (or greater), things become a little lessstraightforward.

If you have a good picture in your mind of the shape of the two curves between -360 degrees and +360 degrees, you may be able to avoid errors once you start working on physics problems that involve angles outsidethe range of 0 to 90 degrees.

Questions & Answers

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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