One of the branches of mathematics that students cope with the most difficulties is trigonometry. Not surprisingly: in order to freely master this area of knowledge, spatial thinking is required, the ability to find sines, cosines, tangents, cotangents by formulas, simplify expressions, and be able to use the number pi in calculations. In addition, you need to be able to apply trigonometry in proving theorems, and this requires either developed mathematical memory or the ability to derive complex logical chains.
The origins of trigonometry
Familiarity with this science should begin with the definition of the sine, cosine and tangent of the angle, but first you need to figure out what trigonometry does.
Historically, the main object of study in this section of mathematical science has been rectangular triangles. The presence of an angle of 90 degrees makes it possible to carry out various operations that allow you to determine the values of all parameters of the figure in question on two sides and one corner or on two angles and one side. In the past, people noticed this pattern and began to actively use it in the construction of buildings, navigation, in astronomy, and even in art.
First stage
Initially, people talked about the relationship of angles and sides exclusively on the example of rectangular triangles. Then special formulas were discovered that made it possible to expand the boundaries of the use of this branch of mathematics in everyday life.
The study of trigonometry at school today begins with rectangular triangles, after which the knowledge gained is used by students in physics and in the solution of 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 different rules apply, and the sum of the angles in a triangle is always more than 180 degrees. This section is not studied at school, however, 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 marking of the surface will be “arched” in three-dimensional space.
Take the globe and thread. Attach the thread to any two points on the globe so that it is taut. Please note - it has taken the shape of an arc. Spherical geometry deals with such forms, which is used in geodesy, astronomy, and other theoretical and applied fields.
Right triangle
Having learned a little about the methods of using trigonometry, we will return to basic trigonometry to further understand what sine, cosine, tangent are, what calculations can be performed with their help, and what formulas are used.
The first step is to understand the concepts related to a right triangle. First, the hypotenuse is the side that lies opposite a 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 two other sides.
For example, if the 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, which form a right angle, are called legs. In addition, it must be remembered that the sum of the angles in a triangle in a rectangular coordinate system is 180 degrees.
Definition
Finally, with a firm understanding of the geometric base, we can turn to the definition of sine, cosine, and tangent of an angle.
The sine of an angle is the ratio of the opposite leg (i.e., 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 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 unity. 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 the calculations or reasoning. This answer is clearly incorrect.
Finally, the tangent of the angle is the ratio of the opposite side to the adjacent one. The same result will be obtained by dividing the sine by cosine. Look: in accordance with the formula, we divide the length of the side by the hypotenuse, then divide by the length of the second side and multiply by the hypotenuse. Thus, we get the same relation as in the definition of tangent.
Cotangent, respectively, is the ratio of the side adjacent to the corner to the opposite. We get the same result by dividing the unit by the tangent.
So, we examined the definitions of what sine, cosine, tangent and cotangent are, and we can do the formulas.
The simplest formulas
In trigonometry one cannot do without formulas - how to find the 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 the angle is equal to one. This formula is a direct consequence of the Pythagorean theorem, however, it saves time if you need to know the angle, not the side.
Many students cannot remember the second formula, which is also very popular in solving school problems: the sum of one and a 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 cosine. It turns out that a simple mathematical operation makes the trigonometric formula completely unrecognizable. Remember: knowing what sine, cosine, tangent and cotangent are, the transformation rules and a few basic formulas, you can at any time print the required more complex formulas yourself on a piece of paper.
Double Angle Formulas and Addition of Arguments
Two other formulas that need to be learned are related to the values of sine and cosine with the sum and difference of angles. They are presented in the figure below. Please note that in the first case, the sine and cosine are multiplied both times, and in the second, 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 workout, try to get them yourself, taking the angle alpha equal to the angle beta.
Finally, note that the double-angle formulas can be converted to lower the degree of sine, cosine, tangent alpha.
Theorems
The two main theorems in basic trigonometry are the sine theorem and the cosine theorem. Using these theorems, you can easily understand how to find the sine, cosine and tangent, and therefore the area of the figure, and the size of each side, etc.
The sine theorem states that as a result of dividing the length of each side of the triangle by the value of the opposite angle, we get the same number. Moreover, this number will be equal to the 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 the product multiplied by the double cosine of the adjacent angle is subtracted from the sum of the squares of two sides - 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.
Inattention Errors
Even knowing what sine, cosine and tangent are, it is easy to make a mistake due to distracted attention or an error in the simplest calculations. To avoid such errors, we will familiarize ourselves with the most popular of them.
First, ordinary fractions should not be converted to decimal until the final result is obtained - you can leave the answer in the form of an ordinary fraction, unless otherwise specified 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 a root of three or two, because they are found in tasks at every step. The same applies to rounding off ugly numbers.
Further, note that the cosine theorem applies to any triangle, but not the Pythagorean theorem! If you mistakenly forget to subtract the double product of the parties, 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 mistake due to carelessness.
Thirdly, do not confuse the values for angles of 30 and 60 degrees for sines, cosines, tangents, cotangents. Remember these values, because a sine of 30 degrees is equal to a cosine of 60, and vice versa. They are easily confused, as a result of which you will inevitably get an erroneous result.
Application
Many students are in no hurry to begin to study trigonometry, because they do not understand its applied meaning. What is sine, cosine, tangent for an engineer or astronomer? These are the concepts due to which one can calculate the distance to distant stars, predict the meteorite’s fall, send a research probe to another planet. Without them, you cannot build a building, design a car, calculate the load on the surface or the trajectory of the subject. And these are just the most obvious examples! After all, trigonometry in one form or another is used everywhere, from music to medicine.
Finally
So, you know what sine, cosine, tangent are. You can use them in calculations and successfully solve school problems.
The whole essence of trigonometry comes down to the fact that according to the known parameters of the triangle, unknowns must be calculated. There are six of these parameters: the lengths of three sides and the magnitudes of three angles. All the difference in the tasks is that the input data is not the same.
How to find the sine, cosine, tangent based on the known lengths of the legs or hypotenuse, you now know. Since these terms mean nothing more than a relation, but a relation is a fraction, the main goal of a trigonometric problem is to find the roots of an ordinary equation or a system of equations. And here ordinary school math will help you.