Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Precalculus

A | Basic Functions and Identities

PrecalculusA | Basic Functions and Identities

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.
Figure A1
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.
Figure A2
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.
Figure A3

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.
Figure A4
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.
Figure A5
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.
Figure A6
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.
Figure A7

Trigonometric Identities

Pythagorean Identities cos 2 θ+ sin 2 θ=1 1+ tan 2 θ= sec 2 θ 1+ cot 2 θ= csc 2 θ cos 2 θ+ sin 2 θ=1 1+ tan 2 θ= sec 2 θ 1+ cot 2 θ= csc 2 θ
Even-Odd Identities cos(−θ)=cosθ sec(−θ)=secθ sin(−θ)=sinθ tan(−θ)=tanθ csc(−θ)=cscθ cot(−θ)=cotθ cos(−θ)=cosθ sec(−θ)=secθ sin(−θ)=sinθ tan(−θ)=tanθ csc(−θ)=cscθ cot(−θ)=cotθ
Cofunction Identities cosθ=sin( π 2 θ ) sinθ=cos( π 2 θ ) tanθ=cot( π 2 θ ) cotθ=tan( π 2 θ ) secθ=csc( π 2 θ ) cscθ=sec( π 2 θ ) cosθ=sin( π 2 θ ) sinθ=cos( π 2 θ ) tanθ=cot( π 2 θ ) cotθ=tan( π 2 θ ) secθ=csc( π 2 θ ) cscθ=sec( π 2 θ )
Fundamental Identities tanθ= sinθ cosθ secθ= 1 cosθ cscθ= 1 sinθ cotθ= 1 tanθ = cosθ sinθ tanθ= sinθ cosθ secθ= 1 cosθ cscθ= 1 sinθ cotθ= 1 tanθ = cosθ sinθ
Sum and Difference Identities cos(α+β)=cosαcosβsinαsinβ cos(αβ)=cosαcosβ+sinαsinβ sin(α+β)=sinαcosβ+cosαsinβ sin(αβ)=sinαcosβcosαsinβ tan(α+β)= tanα+tanβ 1tanαtanβ tan(αβ)= tanαtanβ 1+tanαtanβ cos(α+β)=cosαcosβsinαsinβ cos(αβ)=cosαcosβ+sinαsinβ sin(α+β)=sinαcosβ+cosαsinβ sin(αβ)=sinαcosβcosαsinβ tan(α+β)= tanα+tanβ 1tanαtanβ tan(αβ)= tanαtanβ 1+tanαtanβ
Double-Angle Formulas sin(2θ)=2sinθcosθ cos(2θ)= cos 2 θ sin 2 θ cos(2θ)=12 sin 2 θ cos(2θ)=2 cos 2 θ1 tan(2θ)= 2tanθ 1 tan 2 θ sin(2θ)=2sinθcosθ cos(2θ)= cos 2 θ sin 2 θ cos(2θ)=12 sin 2 θ cos(2θ)=2 cos 2 θ1 tan(2θ)= 2tanθ 1 tan 2 θ
Half-Angle Formulas sin α 2 =± 1cosα 2 cos α 2 =± 1+cosα 2 tan α 2 =± 1cosα 1+cosα tan α 2 = sinα 1+cosα tan α 2 = 1cosα sinα sin α 2 =± 1cosα 2 cos α 2 =± 1+cosα 2 tan α 2 =± 1cosα 1+cosα tan α 2 = sinα 1+cosα tan α 2 = 1cosα sinα
Reduction Formulas sin 2 θ= 1cos( 2θ ) 2 cos 2 θ= 1+cos( 2θ ) 2 tan 2 θ= 1cos( 2θ ) 1+cos( 2θ ) sin 2 θ= 1cos( 2θ ) 2 cos 2 θ= 1+cos( 2θ ) 2 tan 2 θ= 1cos( 2θ ) 1+cos( 2θ )
Product-to-Sum Formulas cosαcosβ= 1 2 [ cos(αβ)+cos(α+β) ] sinαcosβ= 1 2 [ sin(α+β)+sin(αβ) ] sinαsinβ= 1 2 [ cos(αβ)cos(α+β) ] cosαsinβ= 1 2 [ sin(α+β)sin(αβ) ] cosαcosβ= 1 2 [ cos(αβ)+cos(α+β) ] sinαcosβ= 1 2 [ sin(α+β)+sin(αβ) ] sinαsinβ= 1 2 [ cos(αβ)cos(α+β) ] cosαsinβ= 1 2 [ sin(α+β)sin(αβ) ]
Sum-to-Product Formulas sinα+sinβ=2sin( α+β 2 )cos( αβ 2 ) sinαsinβ=2sin( αβ 2 )cos( α+β 2 ) cosαcosβ=2sin( α+β 2 )sin( αβ 2 ) cosα+cosβ=2cos( α+β 2 )cos( αβ 2 ) sinα+sinβ=2sin( α+β 2 )cos( αβ 2 ) sinαsinβ=2sin( αβ 2 )cos( α+β 2 ) cosαcosβ=2sin( α+β 2 )sin( αβ 2 ) cosα+cosβ=2cos( α+β 2 )cos( αβ 2 )
Law of Sines sinα a = sinβ b = sinγ c a sinα = b sinβ = c sinγ sinα a = sinβ b = sinγ c a sinα = b sinβ = c sinγ
Law of Cosines a 2 = b 2 + c 2 2bccosα b 2 = a 2 + c 2 2accosβ c 2 = a 2 + b 2 2abcosγ a 2 = b 2 + c 2 2bccosα b 2 = a 2 + c 2 2accosβ c 2 = a 2 + b 2 2abcosγ
Table A1
Order a print copy

As an Amazon Associate we earn from qualifying purchases.

Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions
Citation information

© Dec 8, 2021 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.