Skip to Content

A | Basic Functions and Identities

PrecalculusA | Basic Functions and Identities

Preface
1 Functions
2 Linear Functions
3 Polynomial and Rational Functions
4 Exponential and Logarithmic Functions
5 Trigonometric Functions
6 Periodic Functions
7 Trigonometric Identities and Equations
8 Further Applications of Trigonometry
9 Systems of Equations and Inequalities
10 Analytic Geometry
11 Sequences, Probability and Counting Theory
12 Introduction to Calculus
A | Basic Functions and Identities
Answer Key
Index

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	$\begin{array}{l} \cos^{2} θ + \sin^{2} θ = 1 \\ 1 + \tan^{2} θ = \sec^{2} θ \\ 1 + \cot^{2} θ = \csc^{2} θ \end{array}$
Even-Odd Identities	$\begin{array}{l} \cos (- θ) = \cos θ \\ \sec (- θ) = \sec θ \\ \sin (- θ) = - \sin θ \\ \tan (- θ) = - \tan θ \\ \csc (- θ) = - \csc θ \\ \cot (- θ) = - \cot θ \end{array}$
Cofunction Identities	$\begin{array}{l} \cos θ = \sin (\frac{π}{2} - θ) \\ \sin θ = \cos (\frac{π}{2} - θ) \\ \tan θ = \cot (\frac{π}{2} - θ) \\ \cot θ = \tan (\frac{π}{2} - θ) \\ \sec θ = \csc (\frac{π}{2} - θ) \\ \csc θ = \sec (\frac{π}{2} - θ) \end{array}$
Fundamental Identities	$\begin{array}{l} \tan θ = \frac{\sin θ}{\cos θ} \\ \sec θ = \frac{1}{\cos θ} \\ \csc θ = \frac{1}{\sin θ} \\ cot θ = \frac{1}{tan θ} = \frac{cos θ}{sin θ} \end{array}$
Sum and Difference Identities	$\begin{array}{l} \cos (α + β) = \cos α \cos β - \sin α \sin β \\ \cos (α - β) = \cos α \cos β + \sin α \sin β \\ \sin (α + β) = \sin α \cos β + \cos α \sin β \\ \sin (α - β) = \sin α \cos β - \cos α \sin β \\ \tan (α + β) = \frac{\tan α + \tan β}{1 - \tan α \tan β} \\ \tan (α - β) = \frac{\tan α - \tan β}{1 + \tan α \tan β} \end{array}$
Double-Angle Formulas	$\begin{array}{l} \sin (2 θ) = 2 \sin θ \cos θ \\ \cos (2 θ) = \cos^{2} θ - \sin^{2} θ \\ \cos (2 θ) = 1 - 2 \sin^{2} θ \\ \cos (2 θ) = 2 \cos^{2} θ - 1 \\ \tan (2 θ) = \frac{2 \tan θ}{1 - \tan^{2} θ} \end{array}$
Half-Angle Formulas	$\begin{array}{l} \sin \frac{α}{2} = \pm \sqrt{\frac{1 - \cos α}{2}} \\ \cos \frac{α}{2} = \pm \sqrt{\frac{1 + \cos α}{2}} \\ \tan \frac{α}{2} = \pm \sqrt{\frac{1 - \cos α}{1 + \cos α}} \\ \tan \frac{α}{2} = \frac{\sin α}{1 + \cos α} \\ \tan \frac{α}{2} = \frac{1 - \cos α}{\sin α} \end{array}$
Reduction Formulas	$\begin{array}{l} \sin^{2} θ = \frac{1 - \cos (2 θ)}{2} \\ \cos^{2} θ = \frac{1 + \cos (2 θ)}{2} \\ \tan^{2} θ = \frac{1 - \cos (2 θ)}{1 + \cos (2 θ)} \end{array}$
Product-to-Sum Formulas	$\begin{array}{l} \cos α \cos β = \frac{1}{2} [\cos (α - β) + \cos (α + β)] \\ \sin α \cos β = \frac{1}{2} [\sin (α + β) + \sin (α - β)] \\ \sin α \sin β = \frac{1}{2} [\cos (α - β) - \cos (α + β)] \\ \cos α \sin β = \frac{1}{2} [\sin (α + β) - \sin (α - β)] \end{array}$
Sum-to-Product Formulas	$\begin{array}{l} \sin α + \sin β = 2 \sin (\frac{α + β}{2}) \cos (\frac{α - β}{2}) \\ \sin α - \sin β = 2 \sin (\frac{α - β}{2}) \cos (\frac{α + β}{2}) \\ \cos α - \cos β = - 2 \sin (\frac{α + β}{2}) \sin (\frac{α - β}{2}) \\ \cos α + \cos β = 2 \cos (\frac{α + β}{2}) \cos (\frac{α - β}{2}) \end{array}$
Law of Sines	$\begin{array}{l} \frac{\sin α}{a} = \frac{\sin β}{b} = \frac{\sin γ}{c} \\ \frac{a}{\sin α} = \frac{b}{\sin β} = \frac{c}{\sin γ} \end{array}$
Law of Cosines	$\begin{array}{l} a^{2} = b^{2} + c^{2} - 2 b c \cos α \\ b^{2} = a^{2} + c^{2} - 2 a c \cos β \\ c^{2} = a^{2} + b^{2} - 2 a b cos γ \end{array}$

Table A1

Order a print copy

As an Amazon Associate we earn from qualifying purchases.

Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 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

Use the information below to generate a citation. We recommend using a citation tool such as this one.
- Authors: Jay Abramson
- Publisher/website: OpenStax
- Book title: Precalculus
- Publication date: Oct 23, 2014
- Location: Houston, Texas
- Book URL: https://openstax.org/books/precalculus/pages/1-introduction-to-functions
- Section URL: https://openstax.org/books/precalculus/pages/a-basic-functions-and-identities

© Sep 25, 2020 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 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.