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Master the Unit Circle: Essential Trigonometry Concepts

Surds

Evaluating and simplifying radicals

auto-gen

Square and square roots

Square Root of a Perfect Square

Cubic and cube roots

Cube root of a perfect cube

Converting radicals to mixed radicals

Converting radicals to entire radicals

Adding and subtracting surds

Adding and subtracting radicals

Adding radicals

Subtracting radicals

Multiplying and dividing surds

Multiplying and dividing radicals

Dividing radicals

Multiplying radicals

Rationalize the denominator 

Factorisation

Factoring difference of cubes

Factoring sum of cubes

Factorise by taking out the greatest common factor

Factor by taking out the greatest common factor

Factor by taking out GCF

Factorise by grouping

Factor by grouping

Factorising difference of squares: x^2 - y^2

Factoring difference of squares: <katex>x^2 - y^2</katex>

Factoring difference of squares: x^2 - y^2

Factorising trinomials

Factoring trinomials

Factoring Trinomials

Linear Functions

Distance formula: \(d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

Distance formula: <katex>d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}</katex>

Distance formula

Midpoint formula: \(M = ( \frac{x_1+x_2}2  ,\frac{y_1+y_2}2)\)

Midpoint formula

Midpoint formula: <katex>M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)</katex>

Midpoint formula: <katex>M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)</katex>

Gradient equation: \(m = \frac{y_2-y_1}{x_2- x_1}\)

Slope equation: <katex>m = \frac{y_2-y_1}{x_2- x_1}</katex>

Gradient intercept form: y = mx + b

Slope intercept form: y = mx + b

General form: Ax + By + C = 0

Gradient-point form: \(y - y_1 = m (x - x_1)\)

Graphing linear functions using table of values

Graphing linear functions using x- and y-intercepts

Graphing linear functions using various forms

Graphing from slope-intercept form y=mx+b

Graphing linear functions using a single point and gradient

Word problems of graphing linear functions

Rate of change

Parallel and perpendicular lines in linear functions

Applications of linear relations

Quadratic Functions

Introduction to quadratic functions

Transformations of quadratic functions

Quadratic function in general form: y = ax^2 + bx + c

Converting from general to vertex form by completing the square

Shortcut: Vertex formula

Converting general form into vertex form by applying the vertex formula

Vertex Formula Application

Graphing quadratic functions: General form VS. Vertex form

Finding the quadratic functions for given parabolas

Applications of quadratic functions

Quadratic function in vertex form: y = a(x-p)^2 + q

Completing the square

Completing the Square

Quadratic Equations

Solving quadratic equations by factorising

Solving quadratic equations by factoring

Solving quadratic equations by completing the square

Using quadratic formula to solve quadratic equations

Solving a quadratic equation with TWO REAL SOLUTIONS

Solving a quadratic equation with TWO COMPLEX SOLUTIONS

The discriminant: Nature of roots of quadratic equations

Applications of quadratic equations

Inequalities

Compound inequalities

Analyze the Alternate Form of Compound Inequalities: AND

Compound linear inequalities

Evaluate Compound Inequalities: And

Express linear inequalities graphically and algebraically

Graph the Inequality

Linear Inequalities – Word Problem 

Linear inequalities

Solving one-step linear inequalities

One-step Linear Inequalities

One-step linear inequalities

One-step Linear Inequalities Decimal

One-step Linear Inequalities Decimal with Fraction

One-step Linear Inequalities Fraction

Solve One Step Linear Inequalities Find Solution– Word Problem

One-step Linear Inequalities — Word Problem

Solving multi-step linear inequalities

Multi-step Linear Inequalities

Two-step Linear Inequalities

Multi-step linear inequalities

Two-step linear inequalities

Solve Multi-step Linear Inequalities – Word Problem 

Solve Multi Step Linear Inequalities Solve Equation – Word Problem

Solve Multi Step Linear Inequalities Equation – Word Problem

Solving quadratic inequalities

Simultaneous Equations

Solving 3 variable simultaneous equations by substitution

Solving 3 variable systems of equations by substitution

Solving 3 variable simultaneous equations by elimination

Solving 3 variable systems of equations by elimination

Word problems relating 3 variable simultaneous equations

Word problems relating 3 variable systems of equations

Solving 3 variable simultaneous equations (no solution, infinite solutions)

Solving 3 variable systems of equations (no solution, infinite solutions)

System of Equations With No Solution

System of Equations With Infinite Solutions

System of Equations With Infinite Solutions - Extended

Simultaneous linear equations

System of linear equations

Simultaneous linear-quadratic equations

System of linear-quadratic equations

Simultaneous quadratic-quadratic equations

Operations with Algebraic Fractions

Partial fraction decomposition

Solving equations with algebraic fractions

Solving rational equations

Simplifying algebraic fractions and restrictions

Simplifying rational expressions and restrictions

Adding and subtracting algebraic fractions

Adding and subtracting rational expressions

Adding and Subtracting Rational Expressions

Multiplying algebraic fractions

Multiplying rational expressions

Multiplying Rational Expressions

Multiplying Rational Expressions in Factored Form

Dividing algebraic fractions

Dividing rational expressions

Dividing Rational Expressions

Dividing Rational Expressions in Factored Form

Applications of equations with algebraic fractions

Applications of rational equations

Simplifying complex fractions

Algebraic Division

Polynomial long division

Polynomial synthetic division 

Remainder theorem

Finding the Remainder Using Synthetic Division and the Remainder Theorem

Factor theorem

Using the Factor Theorem to Test For Factors of a Polynomial

Functions

Function notation

Adding functions

Subtracting functions

Multiplying functions

Dividing functions

Operations of Functions – In a Nutshell

Operations with functions

Transformations of Functions

Transformations of functions: Horizontal translations

Transformations of functions: Vertical translations

Reflection across the y-axis: y = f(-x)

Reflection across the x-axis: y = -f(x)

Transformations of functions: Horizontal stretches

Transformations of functions: Vertical stretches

Reciprocal Functions

Graphing reciprocals of linear functions

Get better marks with our comprehensive AS level Maths help ? whether it&apos;s for <a href="https://qualifications.pearson.com/en/qualifications/edexcel-a-levels/mathematics-2008.html">Edexcel</a> (Pearson), <a href="https://www.aqa.org.uk/subjects/mathematics/as-and-a-level">AQA</a>, <a href="https://www.ocr.org.uk/qualifications/by-subject/mathematics/">OCR</a> or <a href="http://www.wjec.co.uk/qualifications/mathematics/mathematics-gce-a-as/">WJEC</a>. Our AS Maths tutors have you all covered!

Keeping with the specifications of all the exam boards, our thorough help for the exam includes topics such as Trigonometry, Factorisation, Quadratic functions, Exponential functions, Logarithms, Calculus, Statistics, and a lot more. Learn the concepts with our online video tutorials that show you step-by-step solutions to even the hardest maths problems. Then, strengthen your understanding with tons of as maths practice.

All our lessons are taught by experienced AS Maths teachers. Let&apos;s finish your revision in no time, and ACE that maths exam.

AS-Level Maths (Legacy)

math

<h2>Is AS level Maths Hard?</h2>
As the name suggests, advanced subsidiary (AS) level maths, expands beyond the basic concepts of maths and into more complex or "advanced" areas of mathematics. If the thought of this seems a little daunting to you, you&apos;re not alone. There are many students across the UK that are put off by the complex sounding maths problems and terms like simultaneous equations, algebraic expressions, and more.
From the outside looking in, AS level maths can seem like a hard course of study, but if you have a firm grasp of the basics and you paid attention in secondary school maths, you should be well equipped to tackle even the trickiest of problems within this course.
Even if you struggled in <a href="[[region_textbook.html|{"textbook_key":"uk-year10"}]]">year 10</a> or <a href="[[region_textbook.html|{"textbook_key":"uk-year11"}]]">year 11</a> maths, you could still earn a good grade within AS level maths. All it would take is some organization, studying and an effective revision tool like <a href="[[region_textbook.html|{"textbook_key":"uk-as-level-maths"}]]">StudyPug</a>.
Our platform has a vast collection of video tutorials and revision guides for each topic of AS level maths. Our content will walk you through all the relevant topics that are bound to come up during your course of study, providing AS level maths step-by-step with videos on the following:
<ul type="disc">
<li>Simultaneous Equations</li>
<li>Differentiation</li>
<li>Circle Theorems</li>
<li>Operations with Surds</li>
<li>Law of Indices</li>
<li>Binomial Expansion</li>
<li>And more.</li>
</ul>
If you want to familiarize yourself with the topics you studied in secondary school before you take on an A level course, we have an extensive collection of materials for revision in trigonometry, statistics, integration vectors, and all other topic studied across the GCSE curriculum.
Our AS level Maths tutoring service will show you easy to understand solutions to even the hardest AS level maths questions. Furthermore, the information that we deliver, reflects everything you&apos;d expect to find in modern maths textbooks and covers the content most likely to appear on your AQA, Edexcel, MEI, WJEC or OCR maths exams.
We understand that each student will have their own unique learning style and not every student will come to us at the same skill level. That&apos;s why we offer an "AS level maths for dummies" approach that starts with the basics and assumes no prior knowledge. The beauty of our on-demand video service is in is flexibility. You don&apos;t need to sit through lessons you already know. You can skip them entirely or simply fast-forward to the section you need.
If you <a href="[[signup.html]]">join StudyPug today</a> as a subscriber, you&apos;ll have direct access to all of our GCSE and AS level maths help, giving you unprecedented access to a wealth of quality videos and resources that can supplement your studies and help you find your way to the right maths solutions.

<h2>What is AS level Maths Like?</h2>
A level maths is separated into two variants. An AS Level maths course is one year long, whereas the A level Maths course, is a 2 year program with the AS acting as its first year.
Within the structure of the course, students are given a choice as to what modules they study. The modules are as follows:
<ul type="disc">
<li>C1-4 - Core Maths</li>
<li>S1 & S2 - Statistics</li>
<li>M1 & M2 - Mechanics</li>
<li>D1 & D2 - Decisions</li>
</ul>
Regardless of whether you&apos;re studying AS or A level maths, you will be required to study the C1-4 modules. AS students then pick one further module to study (statistics, mechanics, or decisions) and A level students, will pick two.
As an example, <a href="https://nrich.maths.org">nrich.maths.org</a> has drafted what a typical AS student studying statistics might encounter on their course (see below).

https://dmn92m25mtw4z.cloudfront.net/img/textbook/uk/textbook-AS-maths-legacy.png

Mastering the Unit Circle: Your Gateway to Trigonometry

What You'll Learn

Define the unit circle as a circle with radius 1 centered at the origin

Derive coordinates of points on the unit circle using cosine for x and sine for y

Evaluate trigonometric ratios for special angles (0°, 90°, 180°, 270°, 360°)

Memorize sine and cosine graphs to recall trigonometric values quickly

Apply symmetry and reflection to find coordinates in all four quadrants

What You'll Practice

Finding exact coordinates of points at special angles on the unit circle

Evaluating sine and cosine values for angles like π/6, π/4, π/3, and their multiples

Using reference angles and symmetry to determine coordinates in different quadrants

Converting between degree and radian measures for circle angles

Why This Matters

Before You Start — Make Sure You Can:

This Unit Includes

2 Video lessons

Practice exercises

Learning resources

Skills

Unit Circle

Trigonometry

Special Angles

Sine and Cosine

Coordinates

Reference Angles

Radians