Trigonometry is an important section of algebra and geometry for studying and understanding. Trigonometry owes its origin to astronomy, because it was scientists-astronomers who first began to study various ratios of quantities in a right-angled triangle. The word "trigonometry" itself is of Greek origin and is translated literally as "measuring a triangle." In this article, we will pay particular attention to the issue of the tangent of an angle. This attitude of what to what?
Definition
The geometric meaning of the concept is as follows: in the context of a right-angled triangle, the tangent of an angle is the ratio of the opposite leg to the adjacent leg. Consider this relationship on a specific figure for ease of understanding.
In this triangle, the tangent of the angle alpha is the ratio of C to A. Now consider another acute angle - β (beta). For beta, the tangent of an angle is the ratio of A to C.
Now we turn to the definition of tangent, which has an algebraic meaning, for this we need a unit circle.
In order to mark the numerical value of the tangent in the Cartesian coordinate system, first you need to draw a straight line x = 1, which will be perpendicular to the abscissa axis and parallel to the ordinate axis. After that, we will postpone the alpha angle from the abscissa axis and extend its side until it intersects with the straight line x = 1. The ordinate of the intersection point in a particular situation will be the numerical value of the tangent of the deferred angle.
From the point of view of algebra, the definition of the tangent has the following form: the tangent of an angle is the ratio of the sine of a given angle to its cosine.
How are tangent related to cotangent?
The tangent is the inverse function of the cotangent, which means that: tg = 1 / ctg. Thus, the ratio of tangent to cotangent is equal to unity: tg / ctg = 1.
Is there a tangent to cosine relationship?
There is such an identity that defines the relationship between these two trigonometric functions: 1 + tg 2 = 1 / cos 2 . Let us try to prove this identity by transforming its left side using the algebraic definition of the tangent:
- 1 + tg 2 = 1 + (sin / cos) 2
Now we bring the expression to a common denominator:
Recall the main trigonometric identity and simplify the expression, after which we get:
This fraction is equal to the right side of the original expression, so we have proved the identity.
Conclusion
We hope that in this article we have answered all your questions. Now you can confidently answer the question: tangent is the relation of what to what, and using both an algebraic and geometric approach. Good luck in your further study of mathematics!