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sin π 2

sin π 3

3 2

cos π 2

cos π 3

1 2

sin π 4

cos π 4

2 2

sin π 6

sin π

0

sin 3 π 2

cos π

−1

cos 0

cos π 6

3 2

sin 0

Numeric

For the following exercises, state the reference angle for the given angle.

240°

60°

170°

100°

80°

315°

135°

45°

5 π 4

2 π 3

π 3

5 π 6

11 π 3

π 3

7 π 4

π 8

π 8

For the following exercises, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the angles on the unit circle, use a calculator and round to three decimal places.

225°

300°

60° , Quadrant IV, sin ( 300° ) = 3 2 , cos ( 300° ) = 1 2

320°

135°

45° , Quadrant II, sin ( 135° ) = 2 2 , cos ( 135° ) = 2 2

210°

120°

60° , Quadrant II, sin ( 120° ) = 3 2 , cos ( 120° ) = 1 2

250°

150°

30° , Quadrant II, sin ( 150° ) = 1 2 , cos ( 150° ) = 3 2

5 π 4

7 π 6

π 6 , Quadrant III, sin ( 7 π 6 ) = 1 2 , cos ( 7 π 6 ) = 3 2

5 π 3

3 π 4

π 4 , Quadrant II, sin ( 3 π 4 ) = 2 2 , cos ( 4 π 3 ) = 2 2

4 π 3

2 π 3

π 3 , Quadrant II, sin ( 2 π 3 ) = 3 2 , cos ( 2 π 3 ) = 1 2

5 π 6

7 π 4

π 4 , Quadrant IV, sin ( 7 π 4 ) = 2 2 , cos ( 7 π 4 ) = 2 2

For the following exercises, find the requested value.

If cos ( t ) = 1 7 and t is in the 4 th quadrant, find sin ( t ) .

If cos ( t ) = 2 9 and t is in the 1 st quadrant, find sin ( t ) .

77 9

If sin ( t ) = 3 8 and t is in the 2 nd quadrant, find cos ( t ) .

If sin ( t ) = 1 4 and t is in the 3 rd quadrant, find cos ( t ) .

15 4

Find the coordinates of the point on a circle with radius 15 corresponding to an angle of 220° .

Find the coordinates of the point on a circle with radius 20 corresponding to an angle of 120° .

( 10 , 10 3 )

Find the coordinates of the point on a circle with radius 8 corresponding to an angle of 7 π 4 .

Find the coordinates of the point on a circle with radius 16 corresponding to an angle of 5 π 9 .

( 2.778 , 15.757 )

State the domain of the sine and cosine functions.

State the range of the sine and cosine functions.

[ 1 , 1 ]

Graphical

For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of t  .

Graph of a quarter circle with angles of 0, 30, 45, 60, and 90 degrees inscribed. Equivalence of angles in radians shown. Points along circle are marked.
Graph of circle with angle of t inscribed. Point of (negative square root of 3 over 2, 1/2) is at intersection of terminal side of angle and edge of circle.

sin t = 1 2 , cos t = 3 2

Graph of circle with angle of t inscribed. Point of (1/2, negative square root of 3 over 2) is at intersection of terminal side of angle and edge of circle.
Graph of circle with angle of t inscribed. Point of (negative square root of 2 over 2, negative square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.

sin t = 2 2 , cos t = 2 2

Graph of circle with angle of t inscribed. Point of (1/2, square root of 3 over 2) is at intersection of terminal side of angle and edge of circle.
Graph of circle with angle of t inscribed. Point of (-1/2, square root of 3 over 2) is at intersection of terminal side of angle and edge of circle.

sin t = 3 2 , cos t = 1 2

Graph of circle with angle of t inscribed. Point of (-1/2, negative square root of 3 over 2) is at intersection of terminal side of angle and edge of circle.
Graph of circle with angle of t inscribed. Point of (square root of 2 over 2, negative square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.

sin t = 2 2 , cos t = 2 2

Graph of circle with angle of t inscribed. Point of (1,0) is at intersection of terminal side of angle and edge of circle.
Graph of circle with angle of t inscribed. Point of (-1,0) is at intersection of terminal side of angle and edge of circle.

sin t = 0 , cos t = 1

Graph of circle with angle of t inscribed. Point of (0.111,0.994) is at intersection of terminal side of angle and edge of circle.
Graph of circle with angle of t inscribed. Point of (0.803,-0.596 is at intersection of terminal side of angle and edge of circle.

sin t = 0.596 , cos t = 0.803

Graph of circle with angle of t inscribed. Point of (negative square root of 2 over 2, square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.
Graph of circle with angle of t inscribed. Point of (square root of 3 over 2, 1/2) is at intersection of terminal side of angle and edge of circle.

sin t = 1 2 , cos t = 3 2

Graph of circle with angle of t inscribed. Point of (negative square root of 3 over 2, -1/2) is at intersection of terminal side of angle and edge of circle.
Graph of circle with angle of t inscribed. Point of (square root of 3 over 2, -1/2) is at intersection of terminal side of angle and edge of circle.

sin t = 1 2 , cos t = 3 2

Graph of circle with angle of t inscribed. Point of (0, -1) is at intersection of terminal side of angle and edge of circle.
Graph of circle with angle of t inscribed. Point of (-0.649, 0.761) is at intersection of terminal side of angle and edge of circle.

sin t = 0.761 , cos t = 0.649

Graph of circle with angle of t inscribed. Point of (-0.948, -0.317) is at intersection of terminal side of angle and edge of circle.
Graph of circle with angle of t inscribed. Point of (0, 1) is at intersection of terminal side of angle and edge of circle.

sin t = 1 , cos t = 0

Technology

For the following exercises, use a graphing calculator to evaluate.

sin 5 π 9

cos 5 π 9

−0.1736

sin π 10

cos π 10

0.9511

sin 3 π 4

cos 3 π 4

−0.7071

sin 98°

cos 98°

−0.1392

cos 310°

sin 310°

−0.7660

Extensions

sin ( 11 π 3 ) cos ( 5 π 6 )

sin ( 3 π 4 ) cos ( 5 π 3 )

2 4

sin ( 4 π 3 ) cos ( π 2 )

sin ( 9 π 4 ) cos ( π 6 )

6 4

sin ( π 6 ) cos ( π 3 )

sin ( 7 π 4 ) cos ( 2 π 3 )

2 4

cos ( 5 π 6 ) cos ( 2 π 3 )

cos ( π 3 ) cos ( π 4 )

2 4

sin ( 5 π 4 ) sin ( 11 π 6 )

sin ( π ) sin ( π 6 )

0

Real-world applications

For the following exercises, use this scenario: A child enters a carousel that takes one minute to revolve once around. The child enters at the point ( 0 , 1 ) , that is, on the due north position. Assume the carousel revolves counter clockwise.

What are the coordinates of the child after 45 seconds?

What are the coordinates of the child after 90 seconds?

( 0 , 1 )

What is the coordinates of the child after 125 seconds?

When will the child have coordinates ( 0.707 , –0.707 ) if the ride lasts 6 minutes? (There are multiple answers.)

37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds

When will the child have coordinates ( −0.866 , −0.5 ) if the ride last 6 minutes?

Questions & Answers

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
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
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Essential precalculus, part 2. OpenStax CNX. Aug 20, 2015 Download for free at http://legacy.cnx.org/content/col11845/1.2
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