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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

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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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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