Understanding trig derivatives
If you have got an assignment that requires dealing with trig derivatives, finding the derivative of a trigonometric function or other manipulations with triangles, you are most probably will have to deal with the differentiation of trigonometric functions. This is a mathematical process that deals with all the above-mentioned problems. As a matter of fact, when it comes to trig derivatives, the majority of students get scared even of the mathematic term, if not to mention the very process of trigonometric functions processing, etc. Indeed, this is one of the most difficult and complicated aspects of mathematics that students have to deal with during their studies. It requires spending a lot of time, significant efforts and lots of practice to get essential skills for solving such mathematical problems. On the other hand, practicing trigonometry, as well as other branches of mathematics (particularly, geometry), helps develop and improve a number of important skills, such as logical thinking, analyzing, etc. In this article, you will learn to cope with the above-mentioned problems, and learn some essentials of mathematics, especially geometry and trigonometry.
As a rule, the most common trigonometric functions consist of sin(x), cos(x) and tan(x). Derivatives can be used for different purposes. For example, they are helpful when obtaining useful characteristics concerning graphs (here, we are talking of maximum, minimum, slopes, peaks and so on). Also, you can apply trig derivatives to the complicated equations of the graph without even turning to the help of the graphing calculator. However, obtaining the derivative can never be easy, as long as you need to have some basic knowledge and elementary practice in this regard, as well as be aware of different tricks. Look what essential steps you need to undertake in order to deal with derivatives:
- First of all, you need to learn and understand the derivative notation. There are two different kinds of notations, each os which has different properties and can be used for different purposes. They are: the Leibniz Notation, which is considered to be the most popular, and the Lagrange’s Notation, which is shorter than the previous one, but is more useful, especially when trying to find the derivative as a function.
- The next step is to make sure you know and understand the definition and the meaning of the derivative and what problems it can help solve. The derivative of equations are recognized as the generic equations when looking for slopes of different tangent to a certain graph. This is quite a complicated deal, but you need to enlighten yourself on this issue.
- Remember using explicit differentiation in case if the given equation has y to one side.
- The next step you’re going to undertake is to plug the equation into the equation.
- Once you have got a new equation after plugging one equation into another, you will have to expand the factor out dx from the equation that you have.
- Get familiar with the patterns for deriving equal types of equations and make sure you understand the patterns.
- After that, you will have to implicit differentiation in case if you cannot write your equation with y on the one side only.
- Finally, you need to differentiate, or in other words, to find the derivative. By this, we mean that you need to find the derivative for the both sides of the given equation with respect to x.
These guidelines are useful when looking for the trig derivatives, although they don’t cover all the details that you may face during the process of differentiation. In order to increase your chances to succeed in this uneasy deal, keep reading further and you will find some useful tips in regard to this issue.
Tips for dealing with trig derivatives
There are common tricks that some students use in order to make their studying of trigonometry, geometry or other branch of mathematics much easier. As for the trigonometry, we suggest you following tips that are provided below:
- Before you proceed to solving the assigned trigonometric problem, it would be better if you memorize the essential derivatives and way of manipulating with them.
- If you are facing a rather big trigonometric problem to de solved, don’t ever panic. But instead, you need to divide the overall trigonometric problem into pieces. Thus, you will have several little problems, which would be much more easier to deal with.
- You need to know and understand your calculator very good. Practice with using different functions of your calculator in order to learn its properties and capabilities. It is even more important to be aware of how to use the tangent and all the derivative functions that your calculator has.
- Practice as much as you can. It concerns the product rule. The chain rule, the quotient rule and also, you need to practice the implicit differentiation, as long as they are the hardest matters to deal. If you practice as much as you can, your capabilities will develop and improve until you solve the trigonometric problems just automatically.
Having said that, not only the knowledge of the trigonometric functions, the methodology how to use them and awareness of their properties is important to succeed within studying trigonometry, but also you need to have some theoretical background. To get some of theoretical knowledge in the area of trigonometry, keep reading further.
Trigonometry as a branch of mathematics
Trigonometry is one of the branches of mathematics that deals with studying of properties, specificities and characteristics of triangles, such as the relationships including lengths and angles of different triangles. The term «trigonometry» has originated from the Greek language and can be translated as the «measure». The history of this branch of mathematics has started more than two thousand years ago and originated from the astronomical studies and basic geometry.
When trigonometry appeared on the list of other sciences, it affected the world of science a lot. The astronomers of that time noticed that there were relationships between the sides of a right-angle triangle and the angles of the geometry figure. That relationships are expressed in such a way, that if you know at least the length of only one side of a right-angle triangle, together with the value of at least one angle of the triangle, you can find out the two other angles by means of determining them algorithmically. All these calculations were called later as trigonometric functions and they were exactly the functions that are used even today. They are successfully applied in pure and applied mathematics. Apart from this, trigonometric functions are widely used when dealing with wave equation, they can be applied to comprehend the cyclical phenomena, as well as to the electrical and mechanical engineering, astronomy, biology, acoustics and music and even ecology. In addition, trigonometry is considered to be the foundation of land surveying (it is the technique/science/profession that deals with identifying three-dimensional position of points and determining the angles, as well as distances between them).
Usually trigonometry is associated with triangles, which is right, but only if you mean planar right-angle triangles. Even though trigonometry can still be applied to the non-right-angle triangles, this is a quite rare occurrence, because in most of cases it is much more convenient to divide the non-right-angle triangle into two other triangles that would be the right-angle ones. Thus, you will have to deal with only right-angle triangle anyway. Trigonometry essentials are taught at middle schools, high schools and sometimes in colleges.
Interesting facts from the history of trigonometry
Sumerian scientists are widely known due to numerous discoveries that that have made and due to significant achievements and contributions to the world of science of that time, which resulted in further exploration and research of the world. Among the studies that Sumerian developed, there was trigonometry, as long as they studied a lot the properties of the ratios of the sides of the equal triangles. Their studies, however, didn’t appear to be an organized and systematic method for solving important trigonometric problems, such as measuring angles and sides of triangles.
Apart from Sumerian, Greek mathematicians also made a number of significant contributions to the studies of trigonometry. Such outstanding scientists as Euclid and Archimedes dedicated a lot of their works to the study of chords and their properties, as well as to the circles and inscribed angles in them. In addition, they managed to prove the theorems from which the trigonometric functions that we know and use today originated. It has to be mentioned, however, that their methodology and strategy of presenting these theorems were rather algebraic than geometric.
Generally speaking, trigonometry as we know it today originated in Surya Siddhanta. The properties of modern sine convention were documented for the first time in the fifth century by an outstanding Indian scientist in the field of mathematics and astronomer known as Aryabhata. Later, these works were translated into different languages and the knowledge of trigonometry widespreaded all over the world. Soon, Islamic scientists, particularly mathematicians were applying all the six trigonometric functions, exploring their properties and tabulating their values, as well as using trigonometrical functions in terms of spherical geometry.
Numerous discoveries in the area of trigonometry, and the knowledge of trigonometric functions and their applications reached the countries of Western Europe in the sixteenth and seventeenth centuries, although at that time, mathematicians used only the basic concepts concerning trigonometry without going deeper in that issue.
Where trigonometry is used
There is a great number of applications of trigonometry, as well as of trigonometric functions. Different techniques are applied in the area of various sciences and even to the everyday life. For example, triangulation as one of trigonometry techniques is widely used in astronomy with the purpose of determining distances between the stars and between the Earth and other objects in the space. The technique of triangulation is also used in geography with the purpose of determining distances between different countries or objects on the planet. Also, it is important to mention that such trigonometric functions as sine and cosine play considerable role in when describing light and sound waves.
Areas that take advantages from the usage of trigonometric functions and trigonometry as a whole, include astronomy (first of all, it helps with locating different positions of objects in the space), navigation (in almost all areas of the Earth, including oceans, aircraft, space and so on), the theory of music, optics, acoustics, seismology, electronics, architecture, all branches of natural sciences, economics, crystallography and development of the computer games, mechanical and electrical engineering, geodesy, etc.
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