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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

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.
Widad Reply
can you send the book attached ?
Ariel
?
Ariel
What is economics
Widad Reply
the study of how humans make choices under conditions of scarcity
AI-Robot
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn Reply
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn
what is ecnomics
Jan Reply
this is the study of how the society manages it's scarce resources
Belonwu
what is macroeconomic
John Reply
macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.
husaini
etc
husaini
difference between firm and industry
husaini Reply
what's the difference between a firm and an industry
Abdul
firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅
Abdulraufu
Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?
Toofiq Reply
explain standard reason why economic is a science
innocent Reply
factors influencing supply
Petrus Reply
what is economic.
Milan Reply
scares means__________________ends resources. unlimited
Jan
economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses
Jan
calculate the profit maximizing for demand and supply
Zarshad Reply
Why qualify 28 supplies
Milan
what are explicit costs
Nomsa Reply
out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials
AI-Robot
concepts of supply in microeconomics
David Reply
economic overview notes
Amahle Reply
identify a demand and a supply curve
Salome Reply
i don't know
Parul
there's a difference
Aryan
Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods
Israr
Hi Sir please how do u calculate Cross elastic demand and income elastic demand?
Abari
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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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