Precalculus

Precalculus is a mathematical course or a set of courses that act as a foundation for calculus. The two main topics that are included under precalculus are algebra and trigonometry. Precalculus courses can be structured differently depending upon the type of career path chosen by students. For kids who choose to pursue careers in engineering and medicine, the focus is placed on topics like conic sections, Euclidean vectors, and so on. For careers in fields such as finance precalculus courses focus on topics such as matrices and linear algebra.

In this article, we will take an in-depth look at how a standard precalculus course is structured along with the topics included and certain associated examples.

Precalculus courses act as a prerequisite for calculus and cover advanced mathematical concepts based on quantitative reasoning and functions. Schools usually divide precalculus is two categories - algebra and trigonometry. Before kids can learn how to find the derivatives and antiderivatives of functions they need to understand the fundamental concepts of algebraic functions, transcendental functions, complex numbers, and variable expressions. These act as the basis for precalculus.

Precalculus Definition

Precalculus, as the name suggests, includes courses that need to be studied before learning calculus. Most precalculus courses usually focus on algebraic and trigonometric topics rather than direct calculus concepts. However, if students do not have a deep-seated knowledge of these topics then differentiating and integrating functions will be a very hard task.

Precalculus helps in preparing for higher-grade and college-level calculus courses. The various topics covered under precalculus are as follows:

Complex Numbers

A complex number can be defined as a number that contains a real part and an imaginary part. Such a number is of the form x + iy, where iy is the imaginary number.

The topics covered under this course of precalculus are as follows.

• Complex Numbers
• Imaginary Numbers
• Modulus and Argument of Complex Numbers
• Complex Conjugate

Polynomials

A polynomial is an algebraic expression that consists of variables, coefficients, andconstants that are joined using algebraic operations like addition, multiplication, positive exponentiation, etc.

The list of polynomial topics that fall under precalculus are given below:

• Polynomials
• Types of Polynomials
• Adding and Subtracting Polynomials
• Polynomial Equations

Trigonometry

Trigonometry relates the sides of a right-angled triangle to its angles using trigonometric ratios.

The precalculus trigonometry topics covered are as follows:

• Trigonometry Functions
• Trigonometry Chart
• Trigonometry Table

Functions

Functions are binary relations between two sets where every element of the domain is mapped to exactly one element of the codomain.

The precalculus functions topics are given below:

• Functions
• Relation and Functions
• Composite Functions
• Logarithmic Functions
• Exponential Functions

Sequences/Progressions

Arithmetic, harmonic and geometric sequences are the three types of sequences that fall under a precalculus course.

The important topics covered are listed below.

• Arithmetic Progression
• Geometric Progression
• Harmonic Progression

Conic Sections

Under conic sections, students are introduced to different geometrical figures and their properties that will be used in calculus.

The subtopics that are covered under this are mentioned below.

• Circle
• Parabola
• Hyperbola
• Ellipse

Matrices

A matrix is a rectangular array where data is organized in the form of rows and columns.

The important topics that fall under matrices are as follows:

• Matrix Operations
• Determinant
• Transpose of a Matrix
• Types of a Matrix

Vectors

Vectors are elements that exist in the vector space and represent a quantity having both direction and magnitude.

Given below are the important topics.

• Types of Vectors
• Dot Product
• Cross Product
• Addition of Vectors

Precalculus formulas are indirectly used in finding the derivatives and integrals of functions. They also help in simplifying simple as well as complicated problems in precalculus. Some of the important precalculus formulas are given below:

Complex Numbers Formulas

• i² = -1
• (a + ib) + (c + id) = (a + c) + i(b + d)
• (a + ib) - (c + id) = (a - c) + i(b - d)
• (a + ib) × (a - ib) = a² + b²

Polynomials Formulas

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab+ b²
• (a + b)(a - b)= a² - b²
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac

Trigonometry Formulas

• Pythagoras Theorem: Hypotenuse² = Perpendicular² + Base²
• sin θ = 1 / cose cθ (OR) cosec θ = 1 / si nθ
• cos θ = 1 / sec θ (OR) sec θ = 1 / cos θ
• tan θ = 1 / cot θ (OR) cot θ = 1 / tan θ
• sin (A + B) = sin A cos B + cos A sin B
• sin (A - B) = sin A cos B - cos A sin B
• cos (A + B) = cos A cos B - sin A sin B
• cos (A - B) = cos A cos B + sin A sin B
• sin 2θ = 2 sin θ cos θ
• cos 2θ = 1 - 2 sin²θ

Exponents Formulas

• a^m × aⁿ = a^{m + n}
• a^m / aⁿ = a^{m - n}
• (ab)^x = a^x b^x
• (a^x)^y = a^xy

Sequences Formulas

• Sum of n terms in an arithmetic sequence, S_n = (n/2) [ 2a + (n - 1) d ]
• Sum of 2 terms in an harmonic sequence = 2ab / (a + b)
• Sum of n terms in a geometric sequence, S_n = a (rⁿ - 1) / (r - 1)

Conic Sections Formulas

• Equation of a circle = (x − h)² + (y − k)² = r², with the center at (h, k)
• Equation of a parabola is: y = a(x - h)² + k or x = a(y - k)² + h
• Equation of a hyperbola: x²/a² - y² / b² = 1
• Equation of an ellipse: x²/a² + y² / b² = 1

Matrices Formulas

• A^-1A = I, where A^-1 is the inverse matrix of A and I is the identity matrix.
• C = A + B, where c_ij = a_ij + b_ij, where A, B and C are square matrices.
• C = A - B, where c_ij = a_ij - b_ij
• kA = [ka_ij], where 'k' is a scalar.

Vectors Formulas

• u + v = (u₁+v₁, u₂+v₂, u₃+v₃), where, u = (u₁, u₂, u₃) and v = (v₁, v₂, v₃)
• u - v = (u₁-v₁, u₂-v₂, u₃-v₃)
• |u| = √(u₁² + u₂² + u₃²)

A precalculus cheat sheet can be very important as it gives a complete overview of various vital concepts before an examination. It helps to streamline the process of studying and preparing for a test. Given below are some important concepts and formulas that cover the scope of precalculus.

Slope - The slope of a line can be defined as the gradient of the line that describes its steepness. y = mx + c is the general equation of a straight line, where m is the slope and c is the y-intercept.

Area of a Triangle - Area of a triangle can be defined as the space covered within the boundary of the triangle. The formula to calculate the area of a triangle is \frac{1}{2}\times base\times height.

Sine Function - The sine function can be defined as the ratio of the perpendicular to the hypotenuse of a right-angled triangle. sin θ = P / H.

Cosine Function - The cosine function is the ratio of the base to the hypotenuse. cos θ = B / H.

Tangent Function - Tangent is the ratio of the sine function to the cosine function. tan θ = P / B.

Ellipse - An ellipse is a curve traced by the set of all points in a plane that have a constant sum from two fixed points.

Hyperbola - A hyperbola is a curve traced by the set of all points in a plane that have a constant difference from two fixed points.

Into Functions - A function where at least one element of the domain will not have a pre-image in the domain will be known as an into function.

Square Matrix - A square matrix is one in which the number of rows will be equal to the number of columns. The inverse of a matrix can only be determined for square matrices.

Apart from being the foundation for calculus, there are several areas that see the use of various topics that come under precalculus. Some of the applications of precalculus are listed below:

• Medical fields use precalculus to determine the rate of spread of diseases.
• In architecture, precalculus is used to build robust structures and to ensure their stability.
• Precalculus is used in impact testing for cars to prevent accidents on the road.
• In the field of finance, precalculus is used to build predictive mathematical models.

☛Related Articles:

• Limits
• Differential Equations
• Sequence and Series

Important Notes on Precalculus

• Precalculus comprises the study of topics that are required to learn about calculus.
• Algebra and Trigonometry and the two broad categories of topics that fall under precalculus.
• The topics that fall under precalculus do not focus on direct calculus concepts however, they are needed in order to compute the derivatives and antiderivatives of a function.

What Would Precalculus Focus On?

Precalculus is a course or a group of courses in mathematical education that is a pre-requisite for learning calculus. The coursework focuses on algebra and trigonometry.

What are the Main Topics of Precalculus?

The two broad categories of precalculus are algebra and trigonometry. Further, the subtopics of precalculus are as follows:

• Polynomials
• Conic sections
• Functions
• Matrices
• Vectors
• Series

What are the important Precalculus Formulas?

The important formulas for precalculus are listed below:

• cos (A + B) = cos A cos B - sin A sin B
• (a - b)² = a² - 2ab + b²
• (a + b)(a - b)= a² - b²
• (a + ib) × (a - ib) = a² + b²
• Sum of n terms in an arithmetic sequence, S_n = (n/2) [ 2a + (n - 1) d ]
• Equation of a circle = (x − h)² + (y − k)² = r², with the center at (h, k)
• Equation of a parabola is: y = a(x - h)² + k or x = a(y - k)² + h

How Hard is Precalculus?

Precalculus is a very vast subject that includes many topics. However, by building crystal clear mathematical concepts and regular revision of topics, precalculus will be like a cakewalk.

What Grade Level is Precalculus?

Precalculus is introduced to students throughout their school careers. However, the advanced precalculus concepts are restricted for higher grades such as 11th and 12th.

Is Precalculus Considered a Calculus Class?

Precalculus includes the set of topics that are required before starting a calculus course. Thus, it cannot be considered a calculus class but it is designed to prepare students for calculus concepts.

What are the Applications of Precalculus?

Precalculus is used in many industries such as finance, engineering, medicine, and architecture to create complex mathematical models. In addition to this, calculus is based on the topics covered under precalculus.