Trigonometry is one of the most important sections of the study of algebra and geometry, which must be understood and learned by students. This science originates already in the II-III millennia BC, when the method of measuring the angle using its degree measure was first implemented. Most likely, you are already familiar with the concepts of the sine of an angle, the cosine of an angle and know their tabular values. Also, most likely, you have already met with the concept of the tangent of an angle and its tabular values, respectively.
The new term that we will meet in this article is cotangent.
Definitions
Algebraic Definition:
the cotangent of an angle is the ratio of the cosine of an angle to the sine of an angle.
Geometric Definition:
in a right-angled triangle for one of its acute angles, its cotangent is the ratio of the adjacent leg to the opposite leg.
Let us recall the symbols of cosine and sine that have already been passed on the letter, and also introduce a new written symbol for cotangent:
- sinus - sin;
- cosine - cos;
- cotangent - ctg.
We remind the reader that cotangent is a trigonometric function.
What is the relationship between tangent and cotangent?
Cotangent is the inverse function of tangent. Based on this, it can be noted that the ratio of cotangent to tangent and vice versa is always equal to unity: tg / ctg = 1.
What is the relationship between cotangent and sine?
In addition to the main connection between the cotangent and the sine, through the definition (ctg = cos / sin), using trigonometric transformations, we can derive another formula without the participation of the cosine: ctg2 + 1 = 1 / sin2.
We carry out the proof of the above formula:
- transform the left side using the definition of cotangent: cos2 / sin2 + 1;
- let's sum the sum under one fractional line: (sin2 + cos2) / sin2;
- applying knowledge of the basic trigonometric identity, we obtain: 1 / sin2.
The resulting expression is identically equal to the right side of the formula, as required.
Cotangent Table Values
Like other trigonometric functions, cotangent also has its own tabular values for angles 0, 30, 45, 60, 90 degrees and derivatives from them. Of course, all these values can be calculated through the ratio of the tabular values of the cosine and sine, however, it is much more convenient to remember the value of this function without resorting to calculations every time if necessary.
Conclusion
We hope that this article was useful to you and helped to understand that this is cotangent. If so, then the road is open for you to further explore the wonderful world of trigonometry. And we, in turn, can only wish you good luck in this difficult but exciting way!