Download a PDF of this page here.

Download the Spanish version here.

Essential Concepts

Trigonometric Functions

• Radian measure is defined such that the angle associated with the arc of length 1 on the unit circle has radian measure 1. An angle with a degree measure of 180° has a radian measure of [latex]\pi[/latex] rad.
• For acute angles [latex]\theta[/latex], the values of the trigonometric functions are defined as ratios of two sides of a right triangle in which one of the acute angles is [latex]\theta[/latex].
• For a general angle [latex]\theta[/latex], let [latex](x,y)[/latex] be a point on a circle of radius [latex]r[/latex] corresponding to this angle [latex]\theta[/latex]. The trigonometric functions can be written as ratios involving [latex]x, \, y[/latex], and [latex]r[/latex].
• The trigonometric functions are periodic. The sine, cosine, secant, and cosecant functions have period [latex]2\pi[/latex]. The tangent and cotangent functions have period [latex]\pi[/latex].

Inverse Functions

• For a function to have an inverse, the function must be one-to-one. Given the graph of a function, we can determine whether the function is one-to-one by using the horizontal line test.
• If a function is not one-to-one, we can restrict the domain to a smaller domain where the function is one-to-one and then define the inverse of the function on the smaller domain.
• For a function [latex]f[/latex] and its inverse [latex]f^{-1}, \, f(f^{-1}(x))=x[/latex] for all [latex]x[/latex] in the domain of [latex]f^{-1}[/latex] and [latex]f^{-1}(f(x))=x[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex].
• Since the trigonometric functions are periodic, we need to restrict their domains to define the inverse trigonometric functions.
• The graph of a function [latex]f[/latex] and its inverse [latex]f^{-1}[/latex] are symmetric about the line [latex]y=x[/latex].

Exponential and Logarithmic Functions

• The exponential function [latex]y=b^x[/latex] is increasing if [latex]b>1[/latex] and decreasing if [latex]0
• The logarithmic function [latex]y=\log_b(x)[/latex] is the inverse of [latex]y=b^x[/latex]. Its domain is [latex](0,\infty)[/latex] and its range is [latex](−\infty,\infty)[/latex].
• The natural exponential function is [latex]y=e^x[/latex] and the natural logarithmic function is [latex]y=\ln x=\log_e x[/latex].
• Given an exponential function or logarithmic function in base [latex]a[/latex], we can make a change of base to convert this function to any base [latex]b>0, \, b \ne 1[/latex]. We typically convert to base [latex]e[/latex].
• The hyperbolic functions involve combinations of the exponential functions [latex]e^x[/latex] and [latex]e^{−x}[/latex]. As a result, the inverse hyperbolic functions involve the natural logarithm.

Key Equations

• Generalized sine function
  [latex]f(x)=A\sin(B(x-\alpha))+C[/latex]
• Inverse functions
  [latex]f^{-1}(f(x))=x[/latex]  for all  [latex]x[/latex]  in  [latex]D[/latex], and  [latex]f(f^{-1}(y))=y[/latex]  for all  [latex]y[/latex]  in  [latex]R[/latex].

Glossary

base

the number [latex]b[/latex] in the exponential function [latex]f(x)=b^x[/latex] and the logarithmic function [latex]f(x)=\log_b x[/latex]

exponent

the value [latex]x[/latex] in the expression [latex]b^x[/latex]

horizontal line test

a function [latex]f[/latex] is one-to-one if and only if every horizontal line intersects the graph of [latex]f[/latex], at most, once

hyperbolic functions

the functions denoted [latex]\sinh, \, \cosh, \, \tanh, \, \text{csch}, \, \text{sech}[/latex], and [latex]\coth[/latex], which involve certain combinations of [latex]e^x[/latex] and [latex]e^{−x}[/latex]

inverse function

for a function [latex]f[/latex], the inverse function [latex]f^{-1}[/latex] satisfies [latex]f^{-1}(y)=x[/latex] if [latex]f(x)=y[/latex]

inverse hyperbolic functions

the inverses of the hyperbolic functions where [latex]\cosh[/latex] and [latex]\text{sech}[/latex] are restricted to the domain [latex][0,\infty)[/latex]; each of these functions can be expressed in terms of a composition of the natural logarithm function and an algebraic function

inverse trigonometric functions

the inverses of the trigonometric functions are defined on restricted domains where they are one-to-one functions

natural exponential function

the function [latex]f(x)=e^x[/latex]

natural logarithm

the function [latex]\ln x=\log_e x[/latex]

number e

as [latex]m[/latex] gets larger, the quantity [latex](1+(1/m))^m[/latex] gets closer to some real number; we define that real number to be [latex]e[/latex]; the value of [latex]e[/latex] is approximately 2.718282

one-to-one function

a function [latex]f[/latex] is one-to-one if [latex]f(x_1) \ne f(x_2)[/latex] if [latex]x_1 \ne x_2[/latex]

restricted domain

a subset of the domain of a function [latex]f[/latex]

periodic function

a function is periodic if it has a repeating pattern as the values of [latex]x[/latex] move from left to right

radians

for a circular arc of length [latex]s[/latex] on a circle of radius 1, the radian measure of the associated angle [latex]\theta[/latex] is [latex]s[/latex]

trigonometric functions

functions of an angle defined as ratios of the lengths of the sides of a right triangle

trigonometric identity

an equation involving trigonometric functions that is true for all angles [latex]\theta[/latex] for which the functions in the equation are defined

Study Tips

Degrees versus Radians

• Memorize key angle conversions: [latex]30°, 45°, 60°, 90°, 180°[/latex] and their radian equivalents.
• Visualize radian measures on a unit circle to understand their relationship to [latex]π[/latex].
• When solving problems, check if the angle measure is in degrees or radians before proceeding.
• In calculus and advanced mathematics, assume angles are in radians unless specified otherwise.
• Use exact values with [latex]π[/latex] for precision in radian measures, rather than decimal approximations.

The Six Basic Trigonometric Functions

• Practice identifying the opposite, adjacent, and hypotenuse sides for different angles in right triangles.
• Memorize the definitions of all six functions and their reciprocal relationships.
• Use SOH-CAH-TOA to remember sine, cosine, and tangent ratios.
• When solving problems, draw and label a right triangle diagram if one isn’t provided.

Trigonometric Identities

• Memorize fundamental identities through regular practice and flashcards. It might be helpful to create visual aids or mnemonics to remember relationships between functions.
• Practice deriving less common identities from the fundamental ones.
• When verifying, start with the more complex side of the equation.
• Look for opportunities to factor, square binomials, or use Pythagorean identities.
• In complex problems, try substituting trigonometric expressions with variables to simplify.
• Always verify solutions in trigonometric equations by plugging them back into the original equation.

Graphs and Periods of the Trigonometric Functions

• Memorize the basic shapes and periods of all six trigonometric functions.
• Practice sketching transformed graphs by applying one transformation at a time.
• Remember that [latex]B[/latex] inside the function affects the period inversely (larger [latex]B[/latex], shorter period).
• When analyzing a transformed function, identify each component ([latex]A, B, α, C[/latex]) and its effect.
• Use technology to verify your hand-drawn graphs and build intuition about transformations.

Inverse Functions

• Practice finding inverses algebraically by switching [latex]x[/latex] and [latex]y[/latex], then solving for [latex]y[/latex].
• Visualize inverse functions as reflections over [latex]y = x[/latex] to understand their relationship.
• Remember that not all functions have inverses; only one-to-one functions do. Use the horizontal line test to quickly determine if a function has an inverse.
• When graphing inverses, pay attention to how the domain and range swap.
• Be careful not to confuse [latex]f^{-1}(x)[/latex] with [latex]1/f(x)[/latex].

Finding a Function’s Inverse

• Remember to check if a function is one-to-one before attempting to find its inverse.
• When swapping x and y, be careful to replace all instances of the variable.
• Always verify your inverse function by composing it with the original function.

Graphing Inverse Functions

• When restricting domains, consider the function’s behavior and choose intervals that ensure one-to-one correspondence.
• Remember that different domain restrictions can lead to different inverse functions for the same original function.
• Always verify that your restricted function is one-to-one using the horizontal line test.

Inverse Trigonometric Functions

• Memorize the domains and ranges of inverse trig functions.
• When composing trig and inverse trig functions, pay attention to domain restrictions.
• Use the unit circle to understand the relationships between trig functions and their inverses.

Exponential Functions

• Memorize the laws of exponents and practice applying them to simplify expressions.
• Compare exponential and power functions graphically to understand their differences.
• When simplifying complex exponential expressions, break them down step-by-step.
• Remember that negative exponents in the denominator can be moved to the numerator with a positive exponent.

Logarithmic Functions

• Memorize the properties of logarithms and practice applying them.
• Use the change-of-base formula to evaluate logarithms with uncommon bases.
• Practice solving equations that combine exponential and logarithmic functions.
• Remember that [latex]\log_b(b^x) = x[/latex] and [latex]b^{\log_b(x)} = x[/latex] for any positive base [latex]b \neq 1[/latex].

Hyperbolic Functions 

• Memorize the basic hyperbolic identities and practice applying them.
• Compare hyperbolic functions to their trigonometric counterparts to understand similarities and differences.
• When working with inverse hyperbolic functions, pay attention to domain restrictions.
• Use the relationship to exponential functions to help evaluate hyperbolic expressions.