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Algebraic operations are critical to many fields of study. In science, they are used to calculate forces to accelerate a car or determine the rate of a chemical reaction. In engineering, they can be used to design safe and efficient systems. They also underpin algorithms and data structures, which are...
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What do I do with the brackets? Brackets are useful to group numbers, pronumerals and operations together as a whole. Whatever is around the brackets affects all the things inside the brackets. Brackets are commonly used to express mathematical formulae. In order to manipulate expressions containing brackets it is necessary...
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How do I deal with fractions involving pronumerals? Adding and subtracting fractions always requires a common denominator (which is the lower half of the fraction). These need to be the same before you can add or subtract. This means converting some of the fractions to forms that are consistent in...
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What is an algebraic fraction? The numerator (top) or denominator (bottom) of a fraction can be in algebraic form involving numbers and variables (represented by pronumerals or letters). We cover the multiplication and division of fractions containing algebraic terms. Simplifying fractions You have probably seen numerical fractions like \(\frac{18}{24}\) before....
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Let's take a look at algebraic fractions where the denominator is a quadratic expression. Such fractions are common in mathematics and engineering. More complicated algebraic fractions can involve polynomials of any order. However in your studies, it is unlikely that you will have to deal with polynomials higher than quadratics....
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Find out how to express an algebraic fraction as a sum of simpler algebraic fractions (partial fractions). The method of partial fractions involves breaking up an algebraic fraction into simpler parts that are added together. This is useful in integration and in finding inverse Laplace and Fourier transforms. Adding fractions...
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Rearranging formulas, also called transposition of formulas, is a necessary skill for most courses. Let's work on some essential skills in manipulating formulas. Introduction Some of the most important equations that we might be required to transpose occur frequently in science, engineering and economics. They are called formulae and give...
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Are you still trying to get that variable on its own from the formula, but it is in a tricky place – or maybe it appears more than once? Here we demonstrate manipulating or rearranging complex formulas, with overviews and practice questions. Let's look at some more complicated formulas where...
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What is algebra? Why are there letters in the equation? This page links to resources that will help you answer these questions. Algebraic expressions involve pronumerals (letters) to represent values. Pronumerals can take many different values. We often need to plug our own values into a given formula for calculating...
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Understanding functions and relations is crucial for exploring more advanced topics in mathematics and their applications in various scientific and engineering domains. In this module, you'll learn about what functions and relations are and you can practice with the provided exercises. Relations A relation is a set of ordered pairs....
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Often the domain of a function will be restricted to a subset of R. This subset is called an interval, and the end points are a and b. Intervals Often the domain of a function will be restricted to a subset of the set of real numbers, \(\mathbb{R}.\) This subset...
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If f-1(x) is the inverse function of a one-to-one function f(x) then f-1(x) is the set of ordered pairs obtained by interchanging the first and second elements in each ordered pair. Definition of an inverse function If \(f^{-1}(x)\) is the inverse function of a one-to-one function \(f(x)\) then \(f^{-1}(x)\) is...
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The absolute value of a number x gives a measure of its size or magnitude regardless of whether it is positive or negative. If a number is plotted on a number line then its absolute value can be considered to be the distance from zero. Introduction The absolute value of...
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Functions which have different rules for each subset of the domain are called hybrid functions. Sometimes they are referred to as piecewise defined functions. An example of a hybrid function is: \[\begin{align*} y=f(x) & =\begin{cases} -x, & x\leq-1\\ 1, & -1<x<1\\ x, & x\geq1. \end{cases} \end{align*}\] Note that this hybrid...
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A linear graph is the simplest way of representing data or a functional relationship. Consequently an understanding of linear graphs is an essential requirement for many courses in science, engineering and mathematics. The Cartesian plane The Cartesian plane is defined by a pair of mutually perpendicular coordinate axes. The horizontal...
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A quadratic graph is the graph of a quadratic function. These graphs have applications in a wide range of fields. Keep reading to learn about parabolas and how to sketch them, then test your knowledge with some exercises. A quadratic function has the form \(y=ax^{2}+bx+c\) where \(a\neq0\) . Parabolas The...
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The known graphs of some simple functions and relations can be used to sketch related, but more complicated functions. If you know the graph of a function, you can often transform it to a graph of a more complex but related function. A simple example is the graph of the...
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Functions and graphs provide a way to represent, analyse, and understand relationships between different quantities. If you are looking to improve your knowledge of this area, the linked resources are a great way to start. Plotting points on a graph Graphs usually have points or markers on them. The location...
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The hyperbolic functions are analogous to the circular (trigonometric) functions and are widely used in engineering, science and mathematics. This module introduces hyperbolic functions, their graphs and similarities to the circular functions. Whereas circular functions are defined on a unit circle, the hyperbolic functions are defined on a hyperbola. Hyperbolic...
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The differentiation and integration of hyperbolic functions allow us to understand how these functions change and accumulate values which teaches us the essential tools for analysing complex mathematical and real-world systems. Keep reading to learn about these functions and put your new knowledge to the test with some exercises. See...
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Providing a function is one to one, it is possible to find an inverse function. This module discusses inverse hyperbolic functions, which are used in advanced calculus for integration and in the solution of differential equations. See Hyperbolic functions for a list of definitions. Inverse hyperbolic functions may also be...
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Information about functions and graphs to improve your maths skills in these areas. HF1 Hyperbolic functions This module introduces hyperbolic functions, their graphs and similarities to the circular functions. HF2 Derivatives and integrals of hyperbolic functions The hyperbolic functions are widely used in engineering, science and mathematics. HF3 Inverse hyperbolic...
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Pythagoras’ theorem shows the relationship between the sides of a right-angled triangle. Knowing the length of two sides of a right-angled triangle, the length of the third side can be calculated. This mathematical formula is fundamental for finding lengths and distances that are difficult to physically measure. Right angled triangles...
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Sine, cos and tan can be defined using side lengths of a right-angled triangle. These side lengths are identified as either the hypotenuse or the opposite or adjacent sides to the angle. This module shows how to apply trigonometric ratios to find a missing side length, or angle, in a...
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How can we apply trigonometry to triangles that do not possess a right-angle? The sine rule shows that the ratio of the length of a side, to the sine of its opposite angle, will be the same for all three sides. The Sine rule can be used to find angles...
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The cosine rule is a generalisation of Pythagoras’ theorem. If you have any two sides of a triangle, as long as you know the angle between them, you can calculate the length of the third side. The cosine rule can be used to solve non-right triangles. The cosine rule Consider...
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Angles are frequently measured in degrees. However, it is sometimes useful to define angles in terms of the length around the unit circle (a circle of radius = 1). This module introduces radians as a measure of angle. Definition of the radian Though angles have commonly been measured in degrees...
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The trigonometric ratios that have been defined in right-angled triangles can be extended to angles greater than 90 degrees Trigonometric functions such as sin, cos and tan are usually defined as the ratios of sides in a right angled triangle. This module defines the trigonometric functions using angles in a...
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If you know the value of a trigonometric function, how do I find all the possible angles that satisfy this expression? The calculator may only give you one answer to an inverse trig question between 0 and 90 degrees (say InvCos = 40°). The unit circle can help you visualise...
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Both the functions y = sin x and y = cos x have a domain of R and a range of [-1,1]. The graphs of both functions have an amplitude of 1 and a period of 2π radians. The functions \(y=\sin x\) and \(y=\cos x\) have a domain of \(\mathbb{R}\)...
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Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables. They are essential tools for simplifying expressions and solving equations in mathematics. On this page, you'll learn about fundamental identities, double angle formulas, sums and difference, and other trigonometric functions. An algebraic expression...
