PRECALCULUS

UNLOCKED
For Precalculus Part 2 | By Cleofus Balaranjith
Module 1: Trigonometric Identities

Sum and Difference Formulas for Sine and Cosine

The sum and difference formulas for sine and cosine allow us to calculate the sine or cosine of the
sum or difference of two angles.

These formulas are derived from the unit circle and

triangle relationships. Refer to the formulas when
you need to find the sine or cosine of a combined
angle.
By analyzing the coordinates of 𝑃1 ​ and 𝑃2 corresponding to
angles α and β, we can derive that the sine and cosine of α−β
directly result from the sums and differences of the sine and
cosine of α and β.

Sum and Difference Formulas for Tangent

Similar to sine and cosine, the sum and difference formulas for tangent help determine
the tangent of the sum or difference of two angles. These are especially useful when
solving trigonometric equations involving tangent.

The formulas come from the sine and cosine sum and difference formulas, with tangent
being defined as tan 𝜃 = sin 𝜃 Τcos 𝜃. By substituting the sine and cosine formulas into
this definition, you can derive the tangent sum and difference formulas.
Module 2: Trigonometric Formulas

Double-Angle Formulas
They allow you to find the trigonometric identities
for an angle that is double the size of a given
angle. These formulas are extremely useful when
you need to calculate trigonometric values without
using a calculator or when simplifying expressions.

The Double-Angle Formulas for cosine can be

rewritten in terms of either sin(θ) or cos(θ)
depending on what you need. This flexibility
makes it easier to work with different expressions
or to simplify more complex problems.

Half-Angle Formulas
The Half-Angle Formulas
allow you to find the
trigonometric identity for half
of a given angle. These are
particularly useful when you
know the trigonometric
values for a full angle and
need to find them for half
Power-Reduction Formulas that angle.
Power-Reduction Formulas
help in rewriting expressions
that involve squared
trigonometric functions. These
are derived from the Double-
Angle Formulas and are useful
when simplifying expressions or
solving integrals in calculus.
Module 3: Sinusoidal Functions and the Law of Sines

The Sine Function

𝒚 = 𝑨𝒔𝒊𝒏 𝑩𝒙 − 𝑪 + 𝑫 Try it on

Change in A (Amplitude): Adjusts

how tall or short the wave is, making
it stretch or shrink vertically.

Change in B (Frequency): Affects

how many cycles the wave completes
in a given interval, changing the
wave’s width and period.

Change in C (Horizontal Shift/Phase

Shift): Shifts the graph left or right,
determining where the wave starts.

Change in D (Vertical Shift): Moves

the graph up or down, adjusting the
midline of the wave.

A Quick Word on the Cosine Function

The cosine function 𝒚 = 𝑨𝒄𝒐𝒔 𝑩𝒙 − 𝑪 + 𝑫 is nearly identical to the sine
function, with the only difference being a phase shift; it starts at its maximum
𝝅
value rather than at zero. Essentially, 𝒄𝒐𝒔(𝒙) is just 𝒔𝒊𝒏(𝒙 + ), meaning you
𝟐
𝝅
can convert one to the other by shifting the graph horizontally by ​.
𝟐
Module 4: Law of Cosines, Heron’s Formula, and Polar Coordinates

Understanding Polar Coordinates

Polar coordinates represent a point using a
distance r from the origin (pole) and an
angle θ from the positive x-axis. Its like a
Unit circle except the distance r is not
always = one.

Plotting Points Using Polar Coordinates: To plot

a point (r,θ), measure the angle θ from the
polar axis and move a distance r in that
direction.
𝜋
The angle ​ is 90° counterclockwise from the polar axis, with
2
the point 3 units from the pole in that direction.

Handling Negative r Values: A negative

r reverses the direction of the point,
moving it in the opposite direction of
the angle θ.

Importance of Polar Coordinates: Polar coordinates are essential for

representing circular and rotational patterns, making them useful in
fields like physics, engineering, and navigation where angular
measurements are key.
Module 5: Polar Coordinates Continued

Exploring Symmetry in Polar Equations

Types of Symmetry in Polar Equations: Polar equations can show
symmetry with respect to the polar axis, 𝜃 = 𝜋2​, or the pole (origin).

Polar Axis Symmetry: If replacing θ with -θ

gives an equivalent equation, the graph is
symmetric about the polar axis (like the x-axis
in Cartesian coordinates)

Line 𝜽 = 𝝅𝟐 Symmetry: If replacing r

with -r and θ with -θ gives the same Pole Symmetry: If replacing r with -r
equation, the graph is symmetric results in an equivalent equation, the
about the line 𝜃 = 𝜋2. graph is symmetric about the pole (like
the origin in Cartesian coordinates).

symmetry
Module 6: Parametric Equations

Parameterizing Curves
What is Parameterizing? It’s breaking down a
curve into equations that give us more info,
like direction and motion.
Why Use Parameters? Some curves (like circles)
can’t be fully captured as regular functions.
Parametric equations help show the full picture.

How It Works: Instead of just one equation, you now have two, one for x(t) and
one for y(t). You can track how both x and y change as t moves along.
Example of Parameterizing: If 𝑦 = 𝑥 2 − 1 and 𝑥 𝑡 = 𝑡, then 𝑦 𝑡 = 𝑡 2 − 1.

2 different equations same outcome

Module 7: Vectors and Systems of Equations

The Difference Between

𝐮 = 𝒖𝟏 𝒖𝟐 and the Point 𝒖𝟏 , 𝒖𝟐 :
Vectors and points might look similar, but they have different purposes – A
vector 𝐮 = 𝒖𝟏 𝒖𝟐 shows direction and magnitude (a quantity), while a point
𝒖𝟏 , 𝒖𝟐 just marks a position on the plane.

Vectors can move around while keeping the same direction and length, while
points are fixed.

Find the Direction of a Vector: 𝒖𝟏

Direction tells us where the vector points, 𝜽= 𝐭𝐚𝐧−𝟏
which we find using the angle it makes 𝒖𝟐
with the positive x-axis.

Relate to real-life scenarios – think of

direction as the heading of a ship or the
orientation of an arrow in archery.
Module 9: Systems of Linear Equations Continued & Partial Fractions

Simplifying Fractions with Partial Decomposition

What Are Partial Fractions? A
way to break down a complex 𝟓𝒙 − 𝟒 𝟐 𝟑
fraction into simpler fractions +
𝒙𝟐 − 𝒙 − 𝟐 𝒙−𝟐 𝒙+𝟏
(like splitting a large pizza into
smaller slices).

Why Do We Care? Simpler

fractions are easier to work with,
especially in calculus. A
complicated integral becoming
easier to solve after
decomposition.

How to Decompose: Factor the

denominator, set up fractions,
solve for constants, and
combine.

Connection to Adding Fractions:

Decomposition is like reversing
the process of finding a common
denominator.
Module 10: Partial Fractions Continued and Introduction to Matrices

Matrices
Add or subtract corresponding elements from matrices with the
same dimensions. For Example:

Dimension
Mismatch

Multiply rows of the first matrix by columns of the second matrix,

adding up the products.

You can only multiply matrices when the number of columns in the
first matrix equals the number of rows in the second matrix.
Module 11: Matrices Continued

Matrix Inverses and Solving Systems

How do you know if one matrix is the inverse of another matrix?
Two matrices A and B are inverses of each other if multiplying them
results in the identity matrix I. This means A × B = I and B × A = I. The
identity matrix is a special square matrix with 1s on the diagonal and
0s elsewhere.

How can you use matrices to solve systems of linear equations?

To solve a system of linear equations, you can represent the system as a matrix
equation AX=B, where A is the coefficient matrix, X is the column matrix of
variables, and B is the constants matrix. You can solve this using various
methods such as:
Module 12: Conics

Conic Sections
What Are Conic Sections?
Conic sections are curves obtained by
intersecting a plane with a cone.
They’re named for the cone they come
from – think of cutting a cone in different
ways to get different shapes.

There are four main types of conic sections:

Generated Created when Formed when Result from a

when a plane a plane cuts a plane plane cutting
cuts parallel to through a intersects a through both
the generating cone cone at an nappes of a
line of the perpendicular angle, but not cone.
cone. to its axis. perpendicular
to the axis.
Module 13: Conics Continued and Introduction to Sequences

Eccentricity of Conic Sections

The eccentricity e of a conic section is the ratio of the distance from
any point on the conic to its focus, to the perpendicular distance from
that point to the nearest directrix.
Module 14: Arithmetic and Geometric Sequences

Sequences
A sequence is an ordered list of numbers, where each number is called
a term. like 2, 4, 6, 8, 10

Finite vs. Infinite Sequences

Arithmetic Sequence
A sequence where the
difference between
consecutive terms is constant.
If you must subtract to get to
the next term, then the
common difference will be
negative.
Arithmetic Sequence
A sequence where each term is
found by multiplying the
previous term by a constant.
Divide each term by the
previous one; if the result is
always the same, it’s geometric.
Module 15: Counting and Probability

Binomial Mastery
Binomial Coefficient: The binomial coefficient, denoted as 𝑛𝑟 , tells us
how many ways we can choose r items from n items without
considering the order.

Combinations Connection: The binomial coefficient is essentially the

number of combinations or ways to select r items from n items, which
is why it’s also called “n choose r.”

This means each term in the expansion is

found by multiplying the binomial
coefficient 𝑛𝑟 by a raised to a power and
b raised to the remaining power.