Finding the function values for the sine and cosine begins with drawing a unit circle, which is centered at the origin and has a radius of 1 unit.
Using the unit circle, the sine of an angle
$\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ equals the
y -value of the endpoint on the unit circle of an arc of length
$\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ whereas the cosine of an angle
$\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ equals the
x -value of the endpoint. See
[link] .
The sine and cosine values are most directly determined when the corresponding point on the unit circle falls on an axis. See
[link] .
When the sine or cosine is known, we can use the Pythagorean Identity to find the other. The Pythagorean Identity is also useful for determining the sines and cosines of special angles. See
[link] .
Calculators and graphing software are helpful for finding sines and cosines if the proper procedure for entering information is known. See
[link] .
The domain of the sine and cosine functions is all real numbers.
The range of both the sine and cosine functions is
$\text{\hspace{0.17em}}[-1,1].\text{\hspace{0.17em}}$
The sine and cosine of an angle have the same absolute value as the sine and cosine of its reference angle.
The signs of the sine and cosine are determined from the
x - and
y -values in the quadrant of the original angle.
An angle’s reference angle is the size angle,
$\text{\hspace{0.17em}}t,$ formed by the terminal side of the angle
$\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ and the horizontal axis. See
[link] .
Reference angles can be used to find the sine and cosine of the original angle. See
[link] .
Reference angles can also be used to find the coordinates of a point on a circle. See
[link] .
Section exercises
Verbal
Describe the unit circle.
The unit circle is a circle of radius 1 centered at the origin.
What do the
x- and
y- coordinates of the points on the unit circle represent?
Discuss the difference between a coterminal angle and a reference angle.
Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle,
$\text{\hspace{0.17em}}t,$ formed by the terminal side of the angle
$\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ and the horizontal axis.
Explain how the cosine of an angle in the second quadrant differs from the cosine of its reference angle in the unit circle.
Explain how the sine of an angle in the second quadrant differs from the sine of its reference angle in the unit circle.
The sine values are equal.
Algebraic
For the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by
$t$ lies.
$\text{sin}(t)<0\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}\text{cos}(t)<0\text{\hspace{0.17em}}$
$\text{sin}(t)>0\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}\mathrm{cos}(t)>0$
I
$\mathrm{sin}(t)>0$ and
$\text{\hspace{0.17em}}\mathrm{cos}(t)<0$
$\mathrm{sin}(t)<0$ and
$\mathrm{cos}(t)>0$
IV
For the following exercises, find the exact value of each trigonometric function.
Questions & Answers
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
In this morden time nanotechnology used in many field .
1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc
2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc
3- Atomobile -MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it