The solution of many both algebraic and geometric tasks is impossible without using such a trigonometric function as sinus. To find the size of the sinus, you can use both the actual determination of the function and the ratio of the identities of trigonometry, the solution formula, as well as the sinus theorem. With each of these methods in more detail and introduces this article.
Finding the sinus size by definition
The formulation of the term "sine" determines this trigonometric value as the ratio of certain sides of the rectangular triangle - the ratio of the category lying against the desired angle, to the hypotenuse.
Consider Δdfg, ∠dfg \u003d 90 °. Then:
- sIND \u003d FG / DG,
- FG is an opposite cat
- DG - hypotenuse of the presented triangle.
Finding the sion value through the formula of the sinus theorem
This theorem is universal, because Allows you to establish a ratio between the angles and the parties not only a rectangular, then an arbitrary triangle.
Consider ΔLMn
- Mn \u003d L, nl \u003d m, ml \u003d n.
- ∠m \u003d η, ∠n \u003d μ, ∠l \u003d γ.
For an arbitrary triangle ΔLMn, the ratio L / sinl \u003d m / sinm \u003d n / sinn is true - each side of the triangle is proportional to the corner sinus, opposite which it is located.
Describing the radius described near the triangle of the circle through R, the ratio of the sinus theorem is true in the following form:
l / SINL \u003d M / SINM \u003d N / SINN \u003d 2R.
The ratio should:
sinl \u003d L / 2R,
sINM \u003d M / 2R,
sINN \u003d N / 2R.
Finding the sinus size through the triangle area
Before you Δdbc with the parties
DB \u003d C,
BC \u003d D,
DC \u003d b.
To find the triangle area, you can use the ratio S \u003d BC / 2SINB (or S \u003d CD / 2SINB, or S \u003d BD / 2SINC). It follows that:
- sIND \u003d BC / 2S,
- sINB \u003d CD / 2S,
- sINC \u003d BD / 2S.
Finding the sinus size through trigonometry identities
The identity expressions are valid for the angle of any degree.
- cos. 2φ + sin. 2φ \u003d 1 ⇒ sin 2φ \u003d 1 - COS 2φ ⇒ ιsinφι \u003d √ 1 - COS 2φ ⇒ sinφ \u003d ± √ 1 - COS 2φ.
- tgφ \u003d sinφ / cosφ ⇒ sinφ \u003d cosφ * tgφ.
- ctgφ \u003d cosφ / sinφ ⇒ sinφ \u003d cosφ / ctgφ.
- 1 / SIN 2Φ \u003d Ctg. 2Φ + 1 ⇒ Sin 2φ \u003d 1 / (CTG 2φ + 1) ⇒ ιsinφι \u003d 1 / √CTG 2φ + 1 ⇒ sinφ \u003d ± 1 / √CTG 2φ + 1.
Finding the sinus value through the conversion formula
- sin (η + μ) \u003d sinη * cosμ + cosη * sinμ,
- sin (η - μ) \u003d sinη * cosμ - cosη * sinμ,
- sinη + sinμ \u003d 2sin ((η + μ) / 2) * cos ((η - μ) / 2),
- sinη - sinμ \u003d 2cos ((η + μ) / 2) * sin ((η - μ) / 2)
- sinη * sinμ \u003d (cos (η - μ) - cos (η + μ)) / 2,
- sinη \u003d 2tg (η / 2) / (1 + TG 2(η / 2)).
- sin2η \u003d 2sinη * cosη
- sin3η \u003d 3sinη - 4Sin 3η.
Finding the Corner Sine - Table
Taking advantage of the brady's table, you can determine the value of the sinus for each angle in the span of 0 ° to 360 °. Most often, the following table quantities are used in solving the tasks of the school course of the geometry:
- sin0 ° \u003d 0, sin90 ° \u003d 1,
- sin30 ° \u003d. 1/2, sin180 ° \u003d 0,
- sin60 ° \u003d √3 / 2, sin270 ° \u003d -1,
- sin45 ° \u003d √2 / 2, sin360 ° \u003d 0.
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