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0.1 Basic functions and identities

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Graphs of the parent functions

Three graphs side-by-side. From left to right, graph of the identify function, square function, and square root function. All three graphs extend from -4 to 4 on each axis.
Three graphs side-by-side. From left to right, graph of the cubic function, cube root function, and reciprocal function. All three graphs extend from -4 to 4 on each axis.
Three graphs side-by-side. From left to right, graph of the absolute value function, exponential function, and natural logarithm function. All three graphs extend from -4 to 4 on each axis.

Graphs of the trigonometric functions

Three graphs of trigonometric functions side-by-side. From left to right, graph of the sine function, cosine function, and tangent function. Graphs of the sine and cosine functions extend from negative two pi to two pi on the x-axis and two to negative two on the y-axis. Graph of tangent extends from negative pi to pi on the x-axis and four to negative 4 on the y-axis.
Three graphs of trigonometric functions side-by-side. From left to right, graph of the cosecant function, secant function, and cotangent function. Graphs of the cosecant function and secant function extend from negative two pi to two pi on the x-axis and ten to negative ten on the y-axis. Graph of cotangent extends from negative two pi to two pi on the x-axis and twenty-five to negative twenty-five on the y-axis.
Three graphs of trigonometric functions side-by-side. From left to right, graph of the inverse sine function, inverse cosine function, and inverse tangent function. Graphs of the inverse sine and inverse tangent extend from negative pi over two to pi over two on the x-axis and pi over two to negative pi over two on the y-axis. Graph of inverse cosine extends from negative pi over two to pi on the x-axis and pi to negative pi over two on the y-axis.
Three graphs of trigonometric functions side-by-side. From left to right, graph of the inverse cosecant function, inverse secant function, and inverse cotangent function.

Trigonometric identities

Pythagorean Identities cos 2 t + sin 2 t = 1 1 + tan 2 t = sec 2 t 1 + cot 2 t = csc 2 t
Even-Odd Identities cos (− t ) = cos   t sec (− t ) = sec   t sin (− t ) = − sin   t tan (− t ) = − tan   t csc (− t ) = − csc   t cot (− t ) = − cot   t
Cofunction Identities cos   t = sin ( π 2 − t ) sin   t = cos ( π 2 − t ) tan   t = cot ( π 2 − t ) cot   t = tan ( π 2 − t ) sec   t = csc ( π 2 − t ) csc   t = sec ( π 2 − t )
Fundamental Identities tan   t = sin   t cos   t sec   t = 1 cos   t csc   t = 1 sin   t cot   t = 1 tan   t = cos   t sin   t
Sum and Difference Identities cos ( α + β ) = cos   α   cos   β − sin   α   sin   β cos ( α − β ) = cos   α   cos   β + sin   α   sin   β sin ( α + β ) = sin   α   cos   β + cos   α   sin   β sin ( α − β ) = sin   α   cos   β − cos   α   sin   β tan ( α + β ) = tan   α + tan   β 1 − tan   α   tan   β tan ( α − β ) = tan   α − tan   β 1 + tan   α   tan   β
Double-Angle Formulas sin ( 2 θ ) = 2   sin   θ   cos   θ cos ( 2 θ ) = cos 2 θ − sin 2 θ cos ( 2 θ ) = 1 − 2   sin 2 θ cos ( 2 θ ) = 2   cos 2 θ − 1 tan ( 2 θ ) = 2   tan   θ 1 − tan 2 θ
Half-Angle Formulas sin   α 2 = ± 1 − cos   α 2 cos   α 2 = ± 1 + cos   α 2 tan   α 2 = ± 1 − cos   α 1 + cos   α tan   α 2 = sin   α 1 + cos   α tan   α 2 = 1 − cos   α sin   α
Reduction Formulas sin 2 θ = 1 − cos ( 2 θ ) 2 cos 2 θ = 1 + cos ( 2 θ ) 2 tan 2 θ = 1 − cos ( 2 θ ) 1 + cos ( 2 θ )
Product-to-Sum Formulas cos   α   cos   β = 1 2 [ cos ( α − β ) + cos ( α + β ) ] sin   α   cos   β = 1 2 [ sin ( α + β ) + sin ( α − β ) ] sin   α   sin   β = 1 2 [ cos ( α − β ) − cos ( α + β ) ] cos   α   sin   β = 1 2 [ sin ( α + β ) − sin ( α − β ) ]
Sum-to-Product Formulas sin   α + sin   β = 2   sin ( α + β 2 )   cos ( α − β 2 ) sin   α − sin   β = 2   sin ( α − β 2 )   cos ( α + β 2 ) cos   α − cos   β = − 2   sin ( α + β 2 )   sin ( α − β 2 ) cos   α + cos   β = 2   cos ( α + β 2 )   cos ( α − β 2 )
Law of Sines sin   α a = sin   β b = sin   γ c a sin   α = b sin   β = c sin   γ
Law of Cosines a 2 = b 2 + c 2 − 2 b c   cos   α b 2 = a 2 + c 2 − 2 a c   cos   β c 2 = a 2 + b 2 − 2 a b   cos   γ

Questions & Answers

calculate molarity of NaOH solution when 25.0ml of NaOH titrated with 27.2ml of 0.2m H2SO4
Gasin Reply
what's Thermochemistry
rhoda Reply
the study of the heat energy which is associated with chemical reactions
Kaddija
How was CH4 and o2 was able to produce (Co2)and (H2o
Edafe Reply
explain please
Victory
First twenty elements with their valences
Martine Reply
what is chemistry
asue Reply
what is atom
asue
what is the best way to define periodic table for jamb
Damilola Reply
what is the change of matter from one state to another
Elijah Reply
what is isolation of organic compounds
IKyernum Reply
what is atomic radius
ThankGod Reply
Read Chapter 6, section 5
Dr
Read Chapter 6, section 5
Kareem
Atomic radius is the radius of the atom and is also called the orbital radius
Kareem
atomic radius is the distance between the nucleus of an atom and its valence shell
Amos
Read Chapter 6, section 5
paulino
Bohr's model of the theory atom
Ayom Reply
is there a question?
Dr
when a gas is compressed why it becomes hot?
ATOMIC
It has no oxygen then
Goldyei
read the chapter on thermochemistry...the sections on "PV" work and the First Law of Thermodynamics should help..
Dr
Which element react with water
Mukthar Reply
Mgo
Ibeh
an increase in the pressure of a gas results in the decrease of its
Valentina Reply
definition of the periodic table
Cosmos Reply
What is the lkenes
Da Reply
what were atoms composed of?
Moses Reply
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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