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A 23-ft ladder leans against a building so that the angle between the ground and the ladder is 80° . How high does the ladder reach up the side of the building?

22.6506 ft

The angle of elevation to the top of a building in New York is found to be 9 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.

The angle of elevation to the top of a building in Seattle is found to be 2 degrees from the ground at a distance of 2 miles from the base of the building. Using this information, find the height of the building.

368.7633 ft

Assuming that a 370-foot tall giant redwood grows vertically, if I walk a certain distance from the tree and measure the angle of elevation to the top of the tree to be 60° , how far from the base of the tree am I?

Review exercises

Angles

For the following exercises, convert the angle measures to degrees.

π 4  

45°

5 π 3

For the following exercises, convert the angle measures to radians.

-210°

7 π 6

180°

Find the length of an arc in a circle of radius 7 meters subtended by the central angle of 85°.

10.385 meters

Find the area of the sector of a circle with diameter 32 feet and an angle of 3 π 5 radians.

For the following exercises, find the angle between 0° and 360° that is coterminal with the given angle.

420°

60°

80°

For the following exercises, find the angle between 0 and 2 π in radians that is coterminal with the given angle.

20 π 11

2 π 11

14 π 5

For the following exercises, draw the angle provided in standard position on the Cartesian plane.

-210°

A graph of a circle with a negative angle inscribed.

75°

5 π 4

A graph of a circle with an angle inscribed.

π 3

Find the linear speed of a point on the equator of the earth if the earth has a radius of 3,960 miles and the earth rotates on its axis every 24 hours. Express answer in miles per hour.

1036.73 miles per hour

A car wheel with a diameter of 18 inches spins at the rate of 10 revolutions per second. What is the car's speed in miles per hour?

Unit Circle: Sine and Cosine Functions

Find the exact value of sin π 3 .

3 2

Find the exact value of cos π 4 .

Find the exact value of cos π .

–1

State the reference angle for 300° .

State the reference angle for 3 π 4 .

π 4

Compute cosine of 330° .

Compute sine of 5 π 4 .

2 2

State the domain of the sine and cosine functions.

State the range of the sine and cosine functions.

[ 1 , 1 ]

The Other Trigonometric Functions

For the following exercises, find the exact value of the given expression.

cos π 6

tan π 4

1

csc π 3

sec π 4

2

For the following exercises, use reference angles to evaluate the given expression.

sec 11 π 3

sec 315°

2

If sec ( t ) = 2.5 , what is the sec ( t ) ?

If tan ( t ) = 0.6 , what is the tan ( t ) ?

0.6

If tan ( t ) = 1 3 , find tan ( t π ) .

If cos ( t ) = 2 2 , find sin ( t + 2 π ) .

2 2 or 2 2

Which trigonometric functions are even?

Which trigonometric functions are odd?

sine, cosecant, tangent, cotangent

Right Triangle Trigonometry

For the following exercises, use side lengths to evaluate.

cos π 4

cot π 3

3 3

tan π 6

cos ( π 2 ) = sin ( __° )

0

csc ( 18 ° ) = sec ( __° )

For the following exercises, use the given information to find the lengths of the other two sides of the right triangle.

cos B = 3 5 , a = 6

b = 8 , c = 10

tan A = 5 9 , b = 6

For the following exercises, use [link] to evaluate each trigonometric function.

A right triangle with side lengths of 11 and 6. Corners A and B are also labeled.

sin A

11 157 157

tan B

For the following exercises, solve for the unknown sides of the given triangle.

A right triangle with corners labeled A, B, and C. Hyptenuse has length of 4 times square root of 2. Other angles measure 45 degrees.

a = 4 ,   b = 4

A right triangle with hypotenuse with length 5, and an angle of 30 degrees.

A 15-ft ladder leans against a building so that the angle between the ground and the ladder is 70° . How high does the ladder reach up the side of the building?

14.0954 ft

The angle of elevation to the top of a building in Baltimore is found to be 4 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.

Practice test

Convert 5 π 6 radians to degrees.

150°

Convert −620° to radians.

Find the length of a circular arc with a radius 12 centimeters subtended by the central angle of 30° .

6.283 centimeters

Find the area of the sector with radius of 8 feet and an angle of 5 π 4 radians.

Find the angle between and 360° that is coterminal with 375° .

15°

Find the angle between 0 and 2 π in radians that is coterminal with 4 π 7 .

Draw the angle 315° in standard position on the Cartesian plane.

A graph of a circle with an angle inscribed.

Draw the angle π 6 in standard position on the Cartesian plane.

A carnival has a Ferris wheel with a diameter of 80 feet. The time for the Ferris wheel to make one revolution is 75 seconds. What is the linear speed in feet per second of a point on the Ferris wheel? What is the angular speed in radians per second?

3.351 feet per second, 2 π 75 radians per second

Find the exact value of sin π 6 .

Compute sine of 240° .

3 2

State the domain of the sine and cosine functions.

State the range of the sine and cosine functions.

[ 1 , 1 ]

Find the exact value of cot π 4 .

Find the exact value of tan π 3 .

3

Use reference angles to evaluate csc 7 π 4 .

Use reference angles to evaluate tan 210 ° .

3 3

If csc t = 0.68 , what is the csc ( t ) ?

If cos t = 3 2 , find cos ( t 2 π ) .

3 2

Which trigonometric functions are even?

Find the missing angle: cos ( π 6 ) = sin ( ___ )

π 3

Find the missing sides of the triangle A B C : sin B = 3 4 , c = 12

Find the missing sides of the triangle.

A right triangle with hyptenuse length of 9 and angle measure of 60 degrees.

a = 9 2 , b = 9 3 2

The angle of elevation to the top of a building in Chicago is found to be 9 degrees from the ground at a distance of 2000 feet from the base of the building. Using this information, find the height of the building.

Questions & Answers

how can chip be made from sand
Eke Reply
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 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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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