Trigonometry is an important part of mathematics, the knowledge of which is widely used in astronomy and in orientation on the ground. This article discusses the definition of sine, cosine, tangent and cotangent as the main trigonometric functions.
What is trigonometry?
This is a science that studies the quantitative properties of triangles, as its name suggests. The process of studying these simple plane figures is carried out using the so-called trigonometric functions.
Even in ancient Babylon and Egypt, people were faced with tasks requiring knowledge of the relationship between the sides and angles of a triangle (for example, during the construction of Egyptian pyramids). However, accurate evidence has not come down to our time that the Babylonians and Egyptians had the necessary mathematical theory to solve problems of this kind.
Trigonometry was developed at the dawn of our era, thanks to the achievements of ancient Greek scientists. The first tables of trigonometric functions were compiled only in the second half of the 15th century.
Right triangle
Before defining a sine and other trigonometric functions, it is necessary to clarify what a right triangle is. He has one of the angles equal to 90 o .
Knowing that the sum of the angles in this figure is 180 o , we can say with confidence that the other two angles in the sum will be 90 o . Moreover, each of them will be less than a right angle.
The sides of a right triangle have names. A segment lying against a right angle is called hypotenuse. The other two sides are the legs.
Introducing trigonometric functions
Now we can define the sine and cosine of the angle, as well as the tangent and cotangent. To do this, construct a circle of unit radius. The figure below shows the triangle ABC, in which the segment AB = 1 - is the hypotenuse (radius of the circle), AC and CB are the legs.
In determining the sine of the angle, it should be said that it is equal to the ratio of the segment BC to the segment AB. It is written as follows: sin (θ) = BC / AB. Since AB = 1, then sin (θ) = BC. In other words, by the sine of the angle of a right-angled triangle, we understand the ratio of the leg that is opposite this angle to the hypotenuse.
Now the definition of the cosine of the angle. This is the attitude of the leg, adjacent to the angle under consideration, to hypotenuse. For the figure above, we have: cos (θ) = AC / AB = AC.
The tangent of an angle is a trigonometric function, which is determined by the ratio of the leg, opposite to this angle, to the adjacent leg. That is, tg (θ) = BC / AC.
Finally, the angle cotangent is the ratio of the adjacent leg to the opposite leg, i.e. ctg (θ) = AC / BC.
Properties of sine and other trigonometric functions
From the introduced definitions of the sine, cosine of the angle, and other functions, several important conclusions about their properties follow:
- First, trigonometric functions are dimensionless quantities.
- Secondly, their value does not depend on the size of the triangle. The latter fact is easy to prove if we turn to the same figure at the top and consider the triangles ABC and AFE. These triangles are similar, because they have a common angle at the vertex A, which means that the following equality holds: BC / AB = FE / AF = sin (θ). Similar equalities can be given for the remaining trigonometric functions.
- Thirdly, any trigonometric function can be expressed using a maximum of two others. This statement is true, since all three sides of the triangle appear in the expressions for two trigonometric functions. For example, tg (θ) = sin (θ) / cos (θ).
Function Frequency
This property was specially made in a separate paragraph of the article, since its consideration deserves special attention.
If you rotate the segment AB (see the figure above) counterclockwise, then point B will run through the entire circle of unit radius. How will the trigonometric functions change?
Consider the sine. According to the definition of the sine of the angle, when θ = 0, then BC = 0, that is, sin (0 o ) = 0. As the angle θ increases, the length of the segment BC increases. In this case, the length AB remains unchanged. This means that sin (θ) is constantly increasing. When the angle θ = 90 o , then BC = AB and sin (90 o ) = 1.
Further rotation of AB counterclockwise reduces the sine to zero at an angle of 180 o (sin (180 o ) = 0).
For angles with values lying between 180 o and 270 o the sine again increases in absolute value, but decreases in absolute values, since the segment BC will lie in the negative region of the ordinate axis. As a result, sin (270 o ) = -1.
Finally, in the 4th quadrant of the circle, when the angle changes from 270 o to 360 o , the absolute value of the sine increases, but its modulus decreases until at 360 o it again becomes zero (sin (360 o ) = sin (0 o ) = 0).
From the analysis it follows that the sine is a periodic function that repeats its values every 360 o . In trigonometry, they are usually used not in degrees, but in radians. Recall that 2 * pi radian is 360 o , where pi = 3.14 is the number pi. The graph of the sin (x) function is shown in the main photo of the article.
If we make similar reasoning, we can show that the cosine is a periodic function with the same period as for the sine, that is, T = 2pi. The tangent and cotangent are also periodic, only for them T = pi.
The table of values of trigonometric functions
This table includes data on the values of sine, cosine, tangent and cotangent for a set of angles. Schoolchildren are forced to learn these meanings by heart.
Currently, thanks to the development of computer science, all programming languages and calculators are equipped with appropriate libraries that allow you to quickly calculate the values of any trigonometric function in a split second.
The table below shows the values for all the named functions of the set of angles. Which are presented, both in degrees and in radians. The letters "ind" mean that the function for this angle is indefinite. In addition to the main four trigonometric functions, the table also shows the secant (sec) and cosecance (csc), which are the inverse cosine and sine, respectively.
Pythagorean theorem and the relationship of sine and cosine
Since the definition of the sine and cosine of an angle is based on the use of a right triangle, these functions can be connected using the Pythagorean theorem.
For the right triangle shown above, we have: sin (α) = b / a and cos (α) = c / a. The Pythagorean theorem is written as follows: c 2 + b 2 = a 2 . If the left and right sides of this expression are divided by a 2 , and then substitute the formulas for sine and cosine, then we get: (sin (α)) 2 + (cos (α)) 2 = 1.