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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 $\pi$ rad.
• For acute angles $\theta$, 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 $\theta$.
• For a general angle $\theta$, let $(x,y)$ be a point on a circle of radius $r$ corresponding to this angle $\theta$. The trigonometric functions can be written as ratios involving $x, \, y$, and $r$.
• The trigonometric functions are periodic. The sine, cosine, secant, and cosecant functions have period $2\pi$. The tangent and cotangent functions have period $\pi$.

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 $f$ and its inverse $f^{-1}, \, f(f^{-1}(x))=x$ for all $x$ in the domain of $f^{-1}$ and $f^{-1}(f(x))=x$ for all $x$ in the domain of $f$.
• Since the trigonometric functions are periodic, we need to restrict their domains to define the inverse trigonometric functions.
• The graph of a function $f$ and its inverse $f^{-1}$ are symmetric about the line $y=x$.

Exponential and Logarithmic Functions

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

Key Equations

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

Glossary

base

the number $b$ in the exponential function $f(x)=b^x$ and the logarithmic function $f(x)=\log_b x$

exponent

the value $x$ in the expression $b^x$

horizontal line test

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

hyperbolic functions

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

inverse function

for a function $f$, the inverse function $f^{-1}$ satisfies $f^{-1}(y)=x$ if $f(x)=y$

inverse hyperbolic functions

the inverses of the hyperbolic functions where $\cosh$ and $\text{sech}$ are restricted to the domain $[0,\infty)$; 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 $f(x)=e^x$

natural logarithm

the function $\ln x=\log_e x$

number e

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

one-to-one function

a function $f$ is one-to-one if $f(x_1) \ne f(x_2)$ if $x_1 \ne x_2$

restricted domain

a subset of the domain of a function $f$

periodic function

a function is periodic if it has a repeating pattern as the values of $x$ move from left to right

radians

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

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 $\theta$ for which the functions in the equation are defined

Study Tips

Degrees versus Radians

• Memorize key angle conversions: $30°, 45°, 60°, 90°, 180°$ and their radian equivalents.
• Visualize radian measures on a unit circle to understand their relationship to $π$.
• 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 $π$ 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 $B$ inside the function affects the period inversely (larger $B$, shorter period).
• When analyzing a transformed function, identify each component ($A, B, α, C$) and its effect.
• Use technology to verify your hand-drawn graphs and build intuition about transformations.

Inverse Functions

• Practice finding inverses algebraically by switching $x$ and $y$, then solving for $y$.
• Visualize inverse functions as reflections over $y = x$ 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 $f^{-1}(x)$ with $1/f(x)$.

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 $\log_b(b^x) = x$ and $b^{\log_b(x)} = x$ for any positive base $b \neq 1$.

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.