One of the branches of mathematics with which schoolchildren cope with the greatest difficulties is trigonometry. No wonder: in order to freely master this area of knowledge, you need spatial thinking, the ability to find sines, cosines, tangents, cotangents using formulas, simplify expressions, and be able to use the number pi in calculations. In addition, you need to be able to apply trigonometry when proving theorems, and this requires either a developed mathematical memory or the ability to deduce complex logical chains.
The Origins of Trigonometry
Introduction to this science should begin with the definition of the sine, cosine and tangent of an angle, but first you need to figure out what trigonometry does in general.
Historically, right triangles have been the main object of research in this section of mathematical science. The presence of an angle of 90 degrees makes it possible to carry out various operations that allow twosides and one corner or two corners and one side to determine the values of all parameters of the figure in question. In the past, people noticed this pattern and began to actively use it in the construction of buildings, navigation, astronomy and even art.
Inception
Initially, people talked about the relationship of angles and sides exclusively on the example of right triangles. Then special formulas were discovered, which made it possible to expand the boundaries of use in everyday life of this section of mathematics.
The study of trigonometry at school today begins with right triangles, after which the knowledge gained is used by students in physics and solving abstract trigonometric equations, work with which begins in high school.
Spherical trigonometry
Later, when science reached the next level of development, formulas with sine, cosine, tangent, cotangent began to be used in spherical geometry, where other rules apply, and the sum of the angles in a triangle is always more than 180 degrees. This section is not studied at school, but it is necessary to know about its existence at least because the earth's surface, and the surface of any other planet, is convex, which means that any surface marking will be "arc-shaped" in three-dimensional space.
Take a globe and a thread. Attach the thread to any two points on the globe so that it is taut. Pay attention - it has acquired the shape of an arc. It deals with such formsspherical geometry used in geodesy, astronomy and other theoretical and applied fields.
Right triangle
Having learned a little about the ways of using trigonometry, let's return to basic trigonometry in order to further understand what sine, cosine, tangent are, what calculations can be performed with their help and what formulas to use.
First of all, you need to understand the concepts related to a right triangle. First, the hypotenuse is the side opposite the 90 degree angle. She is the longest. We remember that according to the Pythagorean theorem, its numerical value is equal to the root of the sum of the squares of the other two sides.
For example, if two sides are 3 and 4 centimeters respectively, the length of the hypotenuse will be 5 centimeters. By the way, the ancient Egyptians knew about this about four and a half thousand years ago.
The two remaining sides that form a right angle are called legs. In addition, we must remember that the sum of the angles in a triangle in a rectangular coordinate system is 180 degrees.
Definition
Finally, having a solid understanding of the geometric base, we can turn to the definition of the sine, cosine and tangent of an angle.
The sine of an angle is the ratio of the opposite leg (that is, the side opposite the desired angle) to the hypotenuse. The cosine of an angle is the ratio of the adjacent leg to the hypotenuse.
Remember that neither sine nor cosine can be greater than one! Why?Because the hypotenuse is by default the longest side of a right triangle. No matter how long the leg is, it will be shorter than the hypotenuse, which means that their ratio will always be less than one. Thus, if you get a sine or cosine with a value greater than 1 in the answer to the problem, look for an error in calculations or reasoning. This answer is clearly wrong.
Finally, the tangent of an angle is the ratio of the opposite side to the adjacent side. The same result will give the division of the sine by the cosine. Look: in accordance with the formula, we divide the length of the side by the hypotenuse, after which we divide by the length of the second side and multiply by the hypotenuse. Thus, we get the same ratio as in the definition of the tangent.
Cotangent, respectively, is the ratio of the side adjacent to the corner to the opposite side. We get the same result by dividing the unit by the tangent.
So, we have considered the definitions of what is sine, cosine, tangent and cotangent, and we can deal with formulas.
Simple formulas
In trigonometry one cannot do without formulas - how to find sine, cosine, tangent, cotangent without them? But this is exactly what is required when solving problems.
The first formula you need to know when starting to study trigonometry says that the sum of the squares of the sine and cosine of an angle is equal to one. This formula is a direct consequence of the Pythagorean theorem, but it saves time if you need to know the value of the angle, not the side.
Many students can't remember the second formula, also verypopular in solving school problems: the sum of one and the square of the tangent of an angle is equal to one divided by the square of the cosine of the angle. Take a closer look: after all, this is the same statement as in the first formula, only both sides of the identity were divided by the square of the cosine. It turns out that a simple mathematical operation makes the trigonometric formula completely unrecognizable. Remember: knowing what a sine, cosine, tangent and cotangent is, the conversion rules and a few basic formulas, you can at any time independently derive the required more complex formulas on a piece of paper.
Double angle formulas and addition of arguments
Two more formulas to learn are related to the sine and cosine values for the sum and difference of angles. They are shown in the figure below. Please note that in the first case, the sine and cosine are multiplied both times, and in the second case, the pairwise product of the sine and cosine is added.
There are also formulas associated with double angle arguments. They are completely derived from the previous ones - as a practice, try to get them yourself, taking the angle of alpha equal to the angle of beta.
Finally, note that the double angle formulas can be converted to reduce the degree of sine, cosine, tangent alpha.
Theorems
The two main theorems in basic trigonometry are the sine theorem and the cosine theorem. With the help of these theorems, you can easily understand how to find the sine, cosine and tangent, and hence the area of \u200b\u200bthe figure, and the magnitudeeach side, etc.
The sine theorem states that as a result of dividing the length of each of the sides of a triangle by the value of the opposite angle, we get the same number. Moreover, this number will be equal to two radii of the circumscribed circle, i.e. the circle containing all points of the given triangle.
The cosine theorem generalizes the Pythagorean theorem, projecting it onto any triangles. It turns out that from the sum of the squares of the two sides, subtract their product, multiplied by the double cosine of the angle adjacent to them - the resulting value will be equal to the square of the third side. Thus, the Pythagorean theorem turns out to be a special case of the cosine theorem.
Mistakes due to inattention
Even knowing what sine, cosine and tangent are, it is easy to make a mistake due to absent-mindedness or an error in the simplest calculations. To avoid such mistakes, let's take a look at the most popular ones.
First of all, don't convert common fractions to decimals before getting the final result - you can leave the answer as a common fraction, unless otherwise stated in the condition. Such a transformation cannot be called a mistake, but it should be remembered that at each stage of the task, new roots may appear, which, according to the author's idea, should be reduced. In this case, you will waste time on unnecessary mathematical operations. This is especially true for values such as the root of three or two, because they occur in tasks at every step. The same goes for rounding."Ugly" numbers.
Next, note that the cosine theorem applies to any triangle, but not the Pythagorean theorem! If you mistakenly forget to subtract twice the product of the sides multiplied by the cosine of the angle between them, you will not only get a completely wrong result, but also demonstrate a complete misunderstanding of the subject. This is worse than a careless mistake.
Third, do not confuse the values for angles of 30 and 60 degrees for sines, cosines, tangents, cotangents. Remember these values, because the sine of 30 degrees is equal to the cosine of 60, and vice versa. It is easy to mix them up, and you will inevitably get an erroneous result.
Application
Many students are in no hurry to start studying trigonometry, because they do not understand its applied meaning. What is sine, cosine, tangent for an engineer or astronomer? These are concepts thanks to which you can calculate the distance to distant stars, predict the fall of a meteorite, send a research probe to another planet. Without them, it is impossible to build a building, design a car, calculate the load on the surface or the trajectory of an object. And these are just the most obvious examples! After all, trigonometry in one form or another is used everywhere, from music to medicine.
In closing
So, you know what sine, cosine, tangent are. You can use them in calculations and successfully solve school problems.
The whole pointtrigonometry is reduced to the fact that according to the known parameters of the triangle, it is necessary to calculate the unknowns. There are six parameters in total: the lengths of three sides and the magnitudes of three angles. The whole difference in the tasks lies in the fact that different input data are given.
How to find the sine, cosine, tangent based on the known lengths of the legs or the hypotenuse, you now know. Since these terms mean nothing more than a ratio, and a ratio is a fraction, the main goal of the trigonometric problem is to find the roots of an ordinary equation or a system of equations. And here the usual school mathematics will help you.
