Trigonometric functions, including the tangent, are most commonly used during the solution of the same names, as well as geometric tasks. What implies the term "tangent" and how to determine it?
Geometric definition of Tangent
To determine the term "tangent", it is necessary to consider the circle, the center of which is located at the point of crossing the axes of the Cartesian coordinate system (x and y axes) - (0.0). The radius of the circle (R) is 1.
- Choose an arbitrary point on this circle and denote it as a (x, y).
- Next, we will spend directly directly under ∠90 ° to OX axis. Received the segments al \u003d y and ol \u003d x.
- Connect T. A (x, y) with the beginning of the coordinates - t. O. The resulting segment AO \u003d R forms a certain angle with the abscissa axis. Denote it as φ.
The tangent of the resulting angle α is the ratio of the ordinate y (Cut Al) to the abscissa x (segment OL)
tgφ \u003d Al / Ol \u003d y / x, with x ≠ 0.
Because Cuts AL and OL are opposite and adjacent, respectively, ΔOAL Cates with ∠loa \u003d 90 °, the concept of tangent determines the relationship between the lengths of the sides of the rectangular triangle.
Tangent angle - ratio of the length of the opposite catech to the length of the side of the adjacent category.
Determination of Tangent through trigonometric identities
Considering a single circle (paragraph 1), it is easy to notice that:
sinφ \u003d al / r \u003d y / 1 \u003d y,
cosφ \u003d ol / r \u003d x / 1 \u003d x.
Previously, it was found that Tgφ \u003d y / x ⇒ Tgφ \u003d sinφ / cosφ.
Based on this, the following identical expression is true:
sinφ. 2.+ cosφ. 2.\u003d 1 ⇒ TGφ \u003d √ (1 / cosφ 2) – 1.
Determination of Tangent through the formula
Returning to a single circle, it is easy to see:
- Take the point B whose coordinates make up, for example (-x, y).
- An angle formed by the segment of OB (R) and the axis of the abscissa is indicated by η.
- Then TGη \u003d Y / (-X) \u003d - (Y / X) \u003d - TGη.
So, the tangent is an odd function.
tG (π / 2 + η) \u003d -CTGη, TG (π + η) \u003d TGη,
tG (π / 2 - η) \u003d Ctgη, TG (π - η) \u003d -tgη,
tG (3π / 2 + η) \u003d -CTGη, TG (2π + η) \u003d TGη,
tG (3π / 2 - η) \u003d Ctgη, Tg (2π - η) \u003d -tgη.
Because Tangent is a function periodic and its period is π (180 °), the above relationships are valid and generally:
tG (πk + η) \u003d TGη
tG (π / 2 + η + πK) \u003d -CTGη, TG (π + η + πk) \u003d TGη,
tG (π / 2 - η + πk) \u003d Ctgη, Tg (π - η + πk) \u003d -tgη,
tG (3π / 2 + η + πk) \u003d -CTGη, TG (2π + η + πK) \u003d TGη,
tG (3π / 2 - η + πk) \u003d Ctgη, Tg (2π - η + πk) \u003d -tgη, where k is any number from the range of valid numbers.
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